The Time Varying Volatility of Macroeconomic Fluctuations

THE TIME VARYING VOLATILITY OF MACROECONOMICFLUCTUATIONS

ALEJANDRO JUSTINIANO AND GIORGIO E. PRIMICERI

Abstract. We investigate the sources of the important shifts in the volatility

of U.S. macroeconomic variables in the postwar period. To this end, we propose

the estimation of DSGE models allowing for time variation in the volatility of

the structural innovations. We apply our estimation strategy to a large-scale

model of the business cycle and �nd that shocks speci�c to the equilibrium

condition of investment account for most of the sharp decline in volatility of

the last two decades.

1. Introduction

It has been well documented that the volatility of output, in�ation and several

other macroeconomic variables of the U.S. economy has exhibited a very high degree

of time variation over the last �fty years (see, for instance, Stock and Watson (2003)

or Sims and Zha (2006)). Perhaps, the most notorious episode of a substantial

volatility shift in recent U.S. economic history is the �Great Moderation,�which

corresponds to the sharp decline in the standard deviation of GDP as well as other

macroeconomic and �nancial variables since the mid 1980s. While signi�cant e¤orts

have been devoted to determine the timing of the Great Moderation (see, among

others, Kim and Nelson (1999), McConnell and Perez-Quiros (2000), Stock and

Watson (2002), Chauvet and Potter (2001), Herrera and Pesavento (2005)), there

is still substantial disagreement about the origin of this common decline in volatility

(see Stock and Watson (2003) for an overview).

In this paper we investigate this issue by estimating a DSGE model in which

the variance of the structural innovations is allowed to change over time. First, we

Date : First draft: July 2005. This version: August 2007.We would like to thank Mark Gertler, Jinill Kim, David Lopez-Salido, Tommaso Monacelli, ErnstSchaumburg, Jim Stock, our discussants Jean Boivin, Hans Genberg, Thomas Lubik, Simon Pot-ter, Juan Rubio-Ramirez, Kevin Salyer, Andrew Scott and Noah Williams, and seminar partic-ipants at several universities and research institutions for comments. We are also grateful toRiccardo Di Cecio for providing some of the investment de�ators data. The views in this paperare solely the responsibility of the authors and should not be interpreted as re�ecting the viewsof the Board of Governors of the Federal Reserve System or any other person associated with theFederal Reserve System.

1

2 ALEJANDRO JUSTINIANO AND GIORGIO E. PRIMICERI

describe an algorithm that allows for simultaneous inference on both the model�s

parameters and the stochastic volatilities. Then, we apply our estimation strat-

egy to a large-scale business cycle model of the U.S. economy, along the lines of

Christiano, Eichenbaum, and Evans (2005) and Smets and Wouters (2003). The

model exhibits a number of real and nominal frictions, and various shocks with a

structural interpretation. The novelty of our set-up is that all of these shocks have

variances that can �uctuate over time.

We believe that this is an interesting innovation because it enables us to identify

the sources of the changes in the volatility of the main macro variables during the

postwar period. Thereafter, we are able to shed light on the nature of the underlying

disturbances responsible for changes in the variability of the U.S. business cycle and,

in particular, the Great Moderation.

The main conclusions we reach in this study are as follows. First, the exoge-

nous structural disturbances hitting the U.S. economy display substantial stochastic

volatility. Nonetheless, the degree of time variation in variances di¤ers considerably

across shocks, being more pronounced for technology disturbances and, particularly,

monetary policy shocks. Consequently, while stochastic volatility is present in all of

the model�s observed endogenous variables, di¤erent series exhibit contrasting pat-

terns of �uctuation in their variances. Hence, it is not surprising that our approach

delivers a substantially better �t of the data compared not only to a homoskedas-

tic model but also relative to a speci�cation that allows for a single jump in the

volatilities.

Second, the decline in the volatility of output, investment, hours and consump-

tion in the early 1980s is largely driven by a change in the variance of the shock

speci�c to the equilibrium condition of investment. This result is robust to various

modi�cations of the baseline model, including those in which we allow for a jump in

all model parameters and, in particular, a switch from passive to active monetary

policy.

Broadly speaking, these shocks to the equilibrium condition of investment cap-

ture innovations speci�c to the return on capital or to the marginal e¢ ciency of the

investment technology. We suggest two particular interpretations of these distur-

bances, which we believe are useful to shed light on the Great Moderation. First,

in our model these disturbances correspond either to investment speci�c technolog-

ical shocks or, equivalently, to shocks to the relative price of investment in terms

of consumption goods. However, our model is not rich enough to exclude some

THE TIME VARYING VOLATILITY OF MACROECONOMIC FLUCTUATIONS 3

alternative interpretations. Therefore, motivated by Bernanke and Gertler (1989)

and Bernanke, Gertler, and Gilchrist (1999), we suggest a second, broader view of

these disturbances as proxying for unmodeled investment �nancial frictions.

We rely on evidence outside our DSGE model to argue for the plausibility of

both interpretations. In particular, in line with the �rst view, we document a

decline in the standard deviation of the relative price of investment as well as of

investment speci�c technology shocks when the latter are identi�ed like in Fisher

(2006). Regarding the second view, and consistent with a recent line of research, we

note that �nancial frictions became less binding at beginning of the 1980s, following

market deregulation and �nancial innovations that allowed �rms and households

increased access to credit markets (Gertler and Lown (1999), Dynan, Elmendorf,

and Sichel (2005), Campbell and Hercowitz (2006)).

More generally, our results suggest that e¤orts to understand the Great Modera-

tion should focus on the dramatic changes in the investment equilibrium condition

that occurred in the early 1980s.

From the methodological standpoint, this paper is related to the statistics lit-

erature on stochastic volatility models (for an overview, see Kim, Shephard, and

Chib (1998)) and on partial non-Gaussian state-space models (Shephard (1994)).

Drawing from this literature, we develop an e¢ cient algorithm, based on Bayesian

Markov chain Monte Carlo (MCMC) methods, for the numerical evaluation of the

posterior of the parameters of interest. Methodologically, the paper closest to ours

is the recent contribution of Fernandez-Villaverde and Rubio-Ramirez (2006) in

nonlinear DSGE estimation. As discussed in section 2, their approach and ours are

complementary as we analyze di¤erent models, using di¤erent solution methods

and estimation algorithms.1

Regarding the application of these techniques, this paper is related to the large

literature using estimated micro-founded models to understand the main sources of

U.S. business cycle �uctuations (see, for instance, Rotemberg and Woodford (1997),

Ireland (2004), Christiano, Eichenbaum, and Evans (2005), Smets and Wouters

(2006), Altig, Christiano, Eichenbaum, and Linde (2005)). However, as mentioned,

we depart from previous work in this area by allowing for time variation in the

volatility of the structural disturbances. In this respect, the paper closest to ours

1 A related analysis is Laforte (2005) who models variances in a small scale macro model as aMarkov switching process.


is perhaps Boivin and Giannoni (2006), although they only allow a one time shift

in parameters and variances and their estimation is tailored to the analysis of

changes in the e¤ectiveness of monetary policy. In addition, the model of Boivin

and Giannoni (2006) abstracts from investment dynamics, which instead turn out

to be crucial in our study of the Great Moderation.

Our approach is also linked to the fairly large literature dealing with the es-

timation of vector autoregressions with heteroskedastic shocks (see, for example,

Bernanke and Mihov (1998), Cogley and Sargent (2005), Sims and Zha (2006),

Primiceri (2005) or Canova, Gambetti, and Pappa (2005)). In contrast to this

strand of work, one advantage of our analysis is that a fully-�edged model provides

an easier interpretation for the structural disturbances hitting the economy.

The paper is organized as follows. Section 2 presents the class of models that

we deal with. Section 3 and 4 illustrate our application to the model of the U.S.

business cycle and sketch the estimation technique. Section 5 and 6 discuss the

estimation results and address the causes of the Great Moderation. Section 7 pro-

vides two interpretations of our results. Section 8 conducts a number of robustness

checks and compares the �t of our baseline stochastic volatility model relative to al-

ternative speci�cations, including some which allow for indeterminacy in the model

solution. Section 9 concludes with some �nal remarks.

2. Stochastic Volatility in DSGE Models

Consider the general class of models summarized by the following system of

equations:

(2.1) Et [f (yt+1; yt; yt�1; �t; �)] = 0,

where yt is a k � 1 vector of endogenous variables, �t is an n � 1 vector of ex-ogenous disturbances, � is a p � 1 vector of structural parameters and Et denotesthe mathematical expectation operator, conditional on the information available at

time t. For example, (2.1) can be thought as a collection of constraints and �rst

order conditions derived from a micro-founded model of consumers and/or �rms

behavior. The novelty of this paper is that the standard deviations of the elements

of �t are allowed to change over time. In particular, we make the assumption that

log �t � �t = �t"t

"t � N(0; In),


where N indicates the normal distribution, In denotes an n � n identity matrixand �t is a diagonal matrix with the n � 1 vector �t of time varying standarddeviations on the main diagonal. Following the stochastic volatility literature (see,

for instance, Kim, Shephard, and Chib (1998)), we assume that each element of �t

evolves (independently) according to the following stochastic processes:

log �i;t = (1� ��i) log �i + ��i log �i;t�1 + �i;t(2.2)

�i;t � N(0; !2i ) i = 1; :::; n.

Observe that modeling the logarithm of �t, as opposed to �t itself, ensures that the

standard deviation of the shocks remains positive at every point in time.

Our objective is to characterize the posterior distribution of the model struc-

tural parameters (�) and the time varying volatility of the shocks (f�tgTt=1). Notethat the model described by (2.1) is in general nonlinear and its solution must be

approximated, as it does not have a closed-form expression. Our solution method

is based on a log-linear approximation of (2.1) around the deterministic steady

state. By deriving the log-linear approximation as a function of the heteroskedastic

shock �t (as opposed to "t), we are able to retain the minimal set of higher order

terms necessary for heteroskedasticity to play a role in the solution, and yet we

can still use standard packages to solve for the resulting linear system of rational

expectations equations.

In a recent related paper, Fernandez-Villaverde and Rubio-Ramirez (2006) deal

with this class of models adopting a second order approximation of the solution

and the particle �lter to evaluate the likelihood function. The two approaches are

complementary. Our approach is only �rst order accurate and neglects the role

of nonlinearities in the model. In those cases where nonlinearities are important,

this approach may overstate the case for heteroskedasticity in the shocks. On the

other hand, our method has the advantage of considerably simplifying the model

solution and inference, allowing us to estimate a rich model of the business cycle

with nominal and real rigidities that is substantially harder to handle with the

methodology of Fernandez-Villaverde and Rubio-Ramirez (2006). In this way, we

can also evaluate the importance of some prominent explanation put forth for the

Great Moderation, in particular the role of monetary policy.


3. The Model

We estimate a relatively large-scale model of the U.S. business cycle, which has

been shown to �t the data fairly well (Del Negro, Schorfheide, Smets, and Wouters

(2006)). The model is based on work by Christiano, Eichenbaum, and Evans (2005)

and Smets and Wouters (2003), to which the reader is referred for additional details.

Our brief illustration of the model follows closely Del Negro, Schorfheide, Smets,

and Wouters (2006).

3.1. Final goods producers. At every point in time t, perfectly competitive �rms

produce the �nal consumption good Yt, using the intermediate goods Yt(i), i 2 [0; 1]and the production technology

Yt =

�Z 1

0

Yt(i)1

1+�p;t di

�1+�p;t.

�p;t follows the exogenous stochastic process

log �p;t = (1� �p) log �p + �p log �p;t�1 + �p;t"p;t,

where "p;t is i:i:d:N(0; 1) and �p;t evolves as in (2.2). Unless otherwise noticed,

this property of a time varying variance applies to all shocks in the model. Pro�t

maximization and zero pro�t condition for the �nal goods producers imply the

following relation between the price of the �nal good (Pt) and the prices of the

intermediate goods (Pt(i), i 2 [0; 1])

Pt =

�Z 1

0

Pt(i)� 1�p;t di

��p;t,

and the following demand function for the intermediate good i:

Yt(i) =

�Pt(i)

Pt

�� 1+�p;t�p;t

Yt.

As a consequence, �p;t will also correspond to the price mark-up over marginal costs

for the �rms producing intermediate goods.

3.2. Intermediate goods producers. A monopolistic �rm produces the inter-

mediate good i using the following production function:

Yt(i) = max�A1��t Kt(i)

�Lt(i)1�� AtF ; 0

,

where, as usual, Kt(i) and Lt(i) denote respectively the capital and labor input

for the production of good i, F represents a �xed cost of production and At is an


exogenous stochastic process capturing the e¤ects of technology. In particular, we

model At as a unit root process, with a growth rate (zt � log At

At�1) that follows

the exogenous process

zt = (1� �z) + �zzt�1 + �z;t"z;t.

As in Calvo (1983), a fraction �p of �rms cannot re-optimize their prices and, as

we allow for indexation, set their prices following the rule

Pt(i) = Pt�1(i)��pt�1�

1��p ,

where �t is de�ned as PtPt�1

and � denotes the steady state value of �t. Subject to

the usual cost minimization condition, re-optimizing �rms choose their price ( ~Pt(i))

by maximizing the present value of future pro�ts

Et

1Xs=0

�sp�s�t+s

nh~Pt(i)

��sj=0�

�pt�1+j�

1��p�iYt+s(i)�

�Wt+sLt+s(i) +R

kt+sKt+s(i)

�o,

where �t+s is the marginal utility of consumption, Wt and Rkt denote respectively

the wage and the rental cost of capital.

3.3. Households. Firms are owned by a continuum of households, indexed by

j 2 [0; 1]. As in Erceg, Henderson, and Levin (2000), while each household is a mo-nopolistic supplier of specialized labor (Lt(j)), a number of �employment agencies�

combines households�specialized labor into labor services available to the interme-

diate �rms

Lt =

�Z 1

0

Lt(j)1

1+�w dj

�1+�w.

Pro�t maximization and a zero pro�t condition for the perfectly competitive em-

ployment agencies imply the following relation between the wage paid by the inter-

mediate �rms and the wage received by the supplier of specialized labor Lt(j)

Wt =

�Z 1

0

Wt(j)� 1�w dj

��w,

and the following labor demand function for labor type j:

Lt(j) =

�Wt(j)

Wt

�� 1+�w�w

Lt.

Each household maximizes the utility function2

Et

1Xs=0

�sbt+s

�log (Ct+s(j)� hCt+s�1(j))� 't+s

Lt+s(j)1+�

1 + �

�,

2 We assume a cashless limit economy as described in Woodford (2003).


where Ct(j) is consumption, h is the �degree�of internal habit formation, 't is a

preference shock that a¤ects the marginal disutility of labor and bt is a �discount

factor�shock a¤ecting both the marginal utility of consumption and the marginal

disutility of labor. These two shocks follow the stochastic processes

log bt = �b log bt�1 + �b;t"b;t

log't = (1� �') log'+ �' log't�1 + �';t"';t.

The household budget constraint is given by

Pt+sCt+s(j) + Pt+sIt+s(j) +Bt+s(j) � Rt+s�1Bt+s�1(j) +Qt+s�1(j) + �t+s +

+Wt+s(j)Lt+s(j) +Rkt+s(j)ut+s(j) �Kt+s�1(j)� Pt+sa(ut+s(j)) �Kt+s�1(j),

where It(j) is investment, Bt(j) denotes holding of government bonds, Rt is the

gross nominal interest rate, Qt(j) is the net cash �ow from participating in state

contingent securities, �t is the per-capita pro�t that households get from owning

the �rms. Households own capital and choose the capital utilization rate which

transforms physical capital ( �Kt(j)) into e¤ective capital

Kt(j) = ut(j) �Kt�1(j),

which is rented to �rms at the rate Rkt (j). The cost of capital utilization is

a(ut+s(j)) per unit of physical capital. Following Altig, Christiano, Eichenbaum,

and Linde (2005), we assume that ut = 1 and a(ut) = 0 in steady state. In our

partially nonlinear approximation of the model solution, only the curvature of the

function a in steady state needs to be speci�ed, � � a00(1)a0(1) . The usual physical

capital accumulation equation is described by

�Kt(j) = (1� �) �Kt�1(j) + �t

�1� S

�It(j)

It�1(j)

��It(j),

where � denotes the depreciation rate and, as in Christiano, Eichenbaum, and

Evans (2005) and Altig, Christiano, Eichenbaum, and Linde (2005), the function

S captures the presence of adjustment costs in investment, with S0 = 0 and S00 >

0. Lucca (2005) shows that this formulation of the adjustment cost function is

equivalent (up to a �rst order approximation of the model) to a generalization of

a time to build assumption. Following Greenwood, Hercowitz, and Krusell (1997)

and Fisher (2006), �t can be interpreted as an investment speci�c technology shock

(or a shock to the production technology of capital goods), as well as a shock to

the relative price of investment in terms of consumption goods. More generally,


�t can be thought of as a disturbance to the equilibrium condition of investment,

given that it a¤ects the return on capital. For space considerations (and somewhat

abusing notation) we label this disturbance as the �investment shock.�While we

will return to the interpretation of this shock in section 7, here we just assume that

it evolves following the exogenous process

log�t = �� log�t�1 + ��;t"�;t.

Following Erceg, Henderson, and Levin (2000), in every period a fraction �w of

households cannot re-optimize their wages and, therefore, set their wages following

the indexation rule

Wt(j) =Wt�1(j) (�t�1ezt�1)

�w (�e )1��w .

The remaining fraction of re-optimizing households set their wages by maximizing

Et

1Xs=0

�sw�sbt+s

��'t+s

Lt+s(j)1+�

1 + �

�,

subject to the labor demand function.

3.4. Monetary and government policies. Monetary policy sets short term nom-

inal interest rates following a Taylor type rule. In particular, the rule allows for

interest rate smoothing and interest rate responses to deviations of in�ation from

the steady state and deviations of output from trend level:

RtR=

�Rt�1R

��R "��t�

�� Yt=AtY=A

��Y #1��Re�R;t"R;t ,

where R is the steady state for the gross nominal interest rate and "R;t is a monetary

policy shock. We also consider, and later discuss, an alternative speci�cation of the

policy rule, in which the monetary authority responds to the output gap, de�ned

as the ratio between output and the level of output that would prevail in a �exible

price and wage economy (see, for instance, Woodford (2003) or Levin, Onatski,

Williams, and Williams (2005)).

Fiscal policy is assumed to be fully Ricardian and public spending is given by

Gt =

�1� 1

gt

�Yt,

where gt is an exogenous disturbance following the stochastic process

log gt = (1� �g) log g + �g log gt�1 + �g;t"g;t.


3.5. Market clearing. The resource constraint is given by

Ct + It +Gt + a(ut) �Kt�1 = Yt,

3.6. Steady state and model solution. Since the technology process At is as-

sumed to have a unit root, consumption, investment, capital, real wages and output

evolve along a stochastic growth path. Once the model is rewritten in terms of de-

trended variables, we can compute the non-stochastic steady state to approximate

the model around it. This procedure delivers a partial nonlinear state space model

of the kind described in Shephard (1994).

We conclude the discussion of the model by specifying the vector of observables,

completing the state space representation of our model:

(3.1) [� log Yt;� logCt;� log It; logLt;� logWt

Pt; �t; Rt],

where � logXt denotes logXt � logXt�1.

4. Inference

4.1. The data. We estimate the model using seven series of U.S. quarterly data,

as in Levin, Onatski, Williams, and Williams (2005) and Del Negro, Schorfheide,

Smets, and Wouters (2006). These series correspond to the vector of observable

variables of our model, reported in section 3.6. The sample for our dataset spans

from 1954QIII up to 2004QIV. All data are extracted from Haver Analytics data-

base (series mnemonics in parenthesis). Following Del Negro, Schorfheide, Smets,

and Wouters (2006), we construct real GDP by diving the nominal series (GDP)

by population (LF and LH) and the GDP De�ator (JGDP). Real series for con-

sumption and investment are obtained in the same manner, although consumption

corresponds only to personal consumption expenditures of non-durables (CN) and

services (CS), while investment is the sum of personal consumption expenditures of

durables (CD) and gross private domestic investment (I). Real wages corresponds

to nominal compensation per hour in the non-farm business sector (LXNFC) di-

vided by the GDP de�ator. Our measure of labor is given by the log of hours of all

persons in the non-farm business sector (HNFBN) divided by population. In�ation

is measured as the quarterly log di¤erence in the GDP de�ator, while for nomi-

nal interest rates we use the e¤ective Federal Funds rate. Di¤erently from Smets

and Wouters (2003), Boivin and Giannoni (2006) or Levin, Onatski, Williams, and

Williams (2005), we do not demean or detrend any series.


4.2. Bayesian inference. Estimation of these models by pure maximum likeli-

hood is extremely challenging. Following a growing recent literature, we adopt a

Bayesian approach to inference, integrating the sample information with weakly

informative priors, which summarize additional information about the parameters

(see, for instance, Smets and Wouters (2003), Del Negro, Schorfheide, Smets, and

Wouters (2006) or Levin, Onatski, Williams, and Williams (2005)). One advantage

of this approach is that it also ameliorates common numerical problems related to

both the �atness of the likelihood function in some regions of the parameter space,

as well as the existence of multiple local maxima.3

Markov chain Monte Carlo (MCMC) methods are used to characterize the poste-

rior distribution of the model�s structural parameters (�), the time varying volatility

of the shocks (f�tgTt=1) and the coe¢ cients of the volatility processes (��; ��; !

2�).

Bayesian methods deal e¢ ciently with the high dimension of the parameter space

and the nonlinearities of the model, by splitting the original estimation problem

into smaller and simpler blocks. In particular, the MCMC algorithm for this paper

is carried out in three steps. First, a Metropolis step is used to draw from the

posterior of the structural coe¢ cients �. Drawing the sequence of time varying

volatilities �T (conditional on �, �, �� and !2) is instead more involved and relies

mostly on the method presented in Kim, Shephard, and Chib (1998). It consists

of transforming a nonlinear and non-Gaussian state space form into a linear and

approximately Gaussian one, which allows the use of simulation smoothers such as

Carter and Kohn (1994) or Durbin and Koopman (2002). Simulating the condi-

tional posterior of��; ��; !

2�is standard, since it is the product of independent

normal-inverse-Gamma distributions. Further details of the estimation are rele-

gated to appendix A, while appendix B discusses checks for the convergence of the

algorithm.

4.3. Priors. As it is customary when taking DSGE models to the data, we �x

a small number of the model parameters to values that are very common in the

existing literature. In particular, we set the steady state share of capital income

(�) to 0:3, the quarterly depreciation rate of capital (�) to 0:025 and the steady

state government spending to GDP ratio to 0:22, which corresponds to the average

share of government spending in total GDP (Gt=Yt) in our sample. Moreover, we

set the autocorrelation of the mark-up shock (�p) to zero. Two reasons motivate

3 See the survey article by An and Schorfheide (2006) for a detailed discussion of these issues.


this choice: �rst, this parameter is weakly identi�ed from the price indexation co-

e¢ cient; second, shutting down this persistence mechanism helps the identi�cation

of indeterminacy in section 8.4 Finally, we set all the autoregressive coe¢ cients of

the log-volatilities, ��s, to 1. The assumption that the volatilities follow geometric

random walk processes serves two main purposes: on the one hand, it helps to

reduce the number of free parameters of the model; on the other hand, it allows us

to focus on lower frequency changes in the volatilities of the endogenous variables

of our macroeconomic model.

The �rst three columns of table 1 report our priors for the remaining parameters

of the model. While most of these priors are relatively disperse and re�ect previous

results in the literature, a few of them deserve some further discussion. First, our

baseline prior distribution assigns zero probability to the indeterminacy region of

the parameter space, although we will relax this assumption in section 8. Second,

for all but one persistence parameters we use a Beta prior, with mean 0:6 and

standard deviation 0:2. The only exception is neutral technology which includes a

unit root already, and for this reason the prior for the autocorrelation of the growth

rate of neutral technology, �z, is centered at 0:4 instead.

Finally, following Del Negro, Schorfheide, Smets, and Wouters (2006), the priors

for the standard deviations of the shocks are fairly disperse and chosen in order

to generate realistic volatilities for the endogenous variables. Notice that these

priors only enter the speci�cation of the model without stochastic volatility that

we estimate simply for comparison.5

The priors on the variance (s2) of the innovations to the log-volatility processes

deserve some comment as well, as these coe¢ cients are new in the DSGE liter-

ature. We chose an inverse-Gamma prior with mean equal to 0:012 for several

reasons. First, assuming that the log-volatilities behave as random walks, this pa-

rameterization implies an average variation of about 25 percent over our sample of

forty years. We regard this as a conservative degree of time variation. Second, in

the context of time varying vector autoregressions, Primiceri (2005) has tested sev-

eral prior speci�cations and concluded that this value attains the highest marginal

likelihood. Nonetheless, we have assessed the sensitivity of our estimates to alter-

native speci�cations of the prior (especially for the variance of the innovation to

4 In an earlier version, Justiniano and Primiceri (2005), we adopted instead a prior favoringhigh autocorrelation in the mark-up shock. Our results are robust to this alternative speci�cation.

5 To be precise, the mean and the variance of these priors are also used to initialize the �lterfor the stochastic volatilities. Alternative values for the initialization leave our results unchanged.


the log-volatilities) and found that these modi�cations had no important in�uence

on the results.

5. Estimation Results

5.1. Parameter estimates. The last three columns of table 1 summarize the pos-

terior distribution of the model coe¢ cients, reporting posterior medians, standard

deviations and 5th and 95th percentiles computed with the draws of our posterior

simulator. All coe¢ cients estimates are fairly tight and seem for the most part in

line with those reported in Del Negro, Schorfheide, Smets, and Wouters (2006) and

Levin, Onatski, Williams, and Williams (2005).

One important exception is the wage stickiness parameter (�w), which is lower

than previous estimates reported in the literature dealing with inference in DSGE

models. In view of the welfare implications of wage rigidity (see, for instance,

Levin, Onatski, Williams, and Williams (2005)) these variations in estimates may

be important, although we do not explore these issues in the current paper. The

inferred median of the Calvo price stickiness parameter (�p) is approximately equal

to 0:9, which is in line with the value found in Smets and Wouters (2003). This

number is higher than recent estimates in micro studies (see, for instance, Bils and

Klenow (2004)), although the presence of indexation mechanisms (which assures

that prices are actually changed in every period) makes the results potentially

more consistent with the micro evidence on the high frequency of price changes.6

For comparison, table 1 also reports posterior medians, standard deviations, 5th

and 95th percentiles of a model estimated with time invariant volatilities. Notice

that most of the coe¢ cient estimates are similar to the stochastic volatility model,

with the exception perhaps of the parameter for the adjustment costs in investment

(S00) that is slightly lower relative to the speci�cation with time varying volatility.

Finally, table 2 shows that the coe¢ cient estimates of the stochastic volatility

model are quite robust to an alternative speci�cation of the monetary policy rule

in which the monetary authority responds to the output gap. From now on, for

space considerations, we only report estimates for our baseline speci�cation in which

the policy authority responds to deviations of output from the neutral technology

trend. Our choice is motivated by the better �t of this speci�cation according to

6 Note that in a previous version of the paper (Justiniano and Primiceri (2005)) we obtained alower estimate of �p as we allowed for autocorrelation in the mark-up shock. As already mentioned,our results are una¤ected by this modi�cation.


the marginal data density (which is roughly 20 log points higher compared to the

gap rule). While a detailed discussion of model �t is presented in section 8.1, it is

important to stress that none of our results below depend on this choice.

5.2. Volatility estimates. Figure 1 presents the plots of the time varying stan-

dard deviations for the seven shocks of our model. Notice that the degree of sto-

chastic volatility varies substantially across disturbances.

The standard deviation of the price mark-up shock (�p;t, �gure 1e) is relatively

stable, while for the two taste shocks ('t and bt, �gures 1f and 1g respectively)

their volatilities exhibit some moderate �uctuations over the sample. In contrast,

the remaining four shocks exhibit a very important degree of time variation.

The exogenous disturbance showing the largest degree of stochastic volatility

is the monetary policy shock ("MPt , �gure 1a), for which the di¤erence between

the lowest and the highest levels of the standard deviation is roughly 500 percent.

Observe that the �Volcker episode�7 is perfectly captured in our estimates, as well

as the reduction in the volatility of monetary policy shocks during the Greenspan

period. In addition, notice that the volatility of monetary policy shocks is relatively

high during the 1970s. This might be at least in part explained by the fact that our

baseline estimation does not allow for breaks in the coe¢ cients of the Taylor rule.

This issue is quite controversial in the existing literature: among others, Clarida,

Gali, and Gertler (2000) have argued in favor of these changes, while Sims and

Zha (2006) have concluded against them. We will return to these ideas in section

8, where we will allow for shifts in the policy rule coe¢ cients and analyze the

robustness of our results to this alternative empirical speci�cation.

Monetary policy shocks are not the only ones exhibiting a clear pattern of �uc-

tuation in their standard deviations. The standard deviation of technology shocks

(zt, �gure 1b) seems to decrease by approximately one third in the second part

of the sample. This is potentially consistent with the observed reduction in the

volatility of GDP in the last two decades, an issue addressed in more detail in the

next section. A similar pattern is observed for the volatilities of the government

spending (gt, �gure 1c) and, particularly, the investment shock (�t, �gure 1d).

One contribution of our analysis is the ability to quantify how the importance

of various shocks has changed over time in generating economic �uctuations. To

7 The �Volcker episode�refers to the high volatility of interest rates in the 1979-1983 period, dueto the monetary targeting regime initiated by chaiman Paul Volcker in response to the dramaticrise in U.S. in�ation in the 1970s.


this end, we analyze the variance decomposition of each series, which will allow

us to later address the causes of the Great Moderation. We perform the variance

decomposition exercise in the following way: for every draw of the parameters and

the volatilities of the exogenous disturbances, we construct the implied variances

of the (endogenous) observable variables, using the state space representation of

the model solution. Then, we re-compute the variances of the observables, by

sequentially setting to zero the volatility of all disturbances but one, for all time

periods. In this way we are able to investigate the contribution of each shock to the

variability of the observable variables. Notice that, since the variances are changing

over time, our variance decomposition is a time varying �object�as well. Due to

space considerations, we do not present the graphs for all variance decompositions.

Instead, we provide a complete characterization of the variance decomposition for

GDP, while for the remaining series we only report the time varying share of the

variance explained by selected shocks.

Figure 2 presents the evolution of the variance shares of GDP growth attributed

to each exogenous disturbance. Consistent with Greenwood, Hercowitz, and Krusell

(2000) and Fisher (2006), the most important shock in explaining the variability

of GDP growth seems to be the investment shock (�gure 2d). Indeed, at least in

the �rst part of the sample, this disturbance explains roughly half of the variance

of GDP growth. Note, however, that the importance of this shock for output

�uctuations declines over time. On average, neutral technology and labor preference

shocks each explain 20 percent of the variance of GDP growth (�gure 2b). Labor

preference shocks play a lesser role in output �uctuations earlier in the sample,

although their importance has increased in the last two decades (�gure 2f). Other

shocks are less central for output.

For the remaining series, �gure 3 plots the time varying variance shares explained

by selected shocks. A major portion of the variance of consumption is explained by

the inter-temporal shock to the discount factor (�gure 3a). Although not crucial

for output, monetary policy and mark-up shocks are each quite important for the

volatility of interest rates (�gure 3b) and in�ation (�gure 3c). Notice, however,

that a major portion of the volatility of in�ation is also explained by the labor

disutility and, particularly, the investment shock (�gure 3d and 3e). Moreover, as

one would expect, the labor disutility and the investment shocks explain most of

the variability of hours (�gure 3g) and investment (�gure 3f) respectively, while


the neutral technology shock accounts for about 20 percent of the variance of real

wages (�gure 3h).8

Figure 4 plots the model�s implied spectral variance decomposition for the level

of output in deviations from the model�s common stochastic trend. We consider

periodicities between 8 and 32 quarters. Figure 4 corroborates the evidence on

the importance of the investment shock, suggesting that this disturbance is crucial

in explaining output �uctuations at business cycle frequencies (�gure 4d). Note,

however, that the labor disutility shock also plays a very important role in this case

(�gure 4f).

6. The Great Moderation

In two very in�uential papers, Kim and Nelson (1999) and McConnell and Perez-

Quiros (2000) drew attention to the dramatic reduction in the volatility of U.S.

GDP, which has characterized the last two decades relative to the pre-1980s pe-

riod. This change seems to be more abrupt than gradual (Kim and Nelson (1999)

and Stock and Watson (2002)) and the break date is estimated to approximately

correspond to 1984. In our sample, the standard deviation of GDP growth over

the 1984-2004 period is almost one half of the standard deviation computed over

the 1955-1983 sample. As mentioned, the literature has labeled this phenomenon

as the Great Moderation.

A number of hypothesis have been put forward to account for this decline in

volatility and exhaustive reviews can be found in Blanchard and Simon (2001) and

Stock and Watson (2002 and 2003). The main explanations of this phenomenon

can be broadly bunched as corresponding to simply good luck, improvements in

the conduct of monetary policy under the Volcker and Greenspan chairmanships

or, alternatively, improvements in inventory management.

With regards to the last explanation, however, several authors have raised doubts

about the improved inventory management hypothesis initially put forward by Mc-

Connell and Perez-Quiros (2000) and Kahn and McConnell (2002). Moreover, our

model is already quite involved and the computational demand of our estimation

algorithm is substantial. For these reasons, our analysis of the Great Moderation

completely abstracts from this channel. We instead refer to the literature for an ex-

haustive review of the reasons why the inventories hypothesis has been questioned,

both from a theoretical (Maccini and Pagan (2003) or Kahn and Thomas (2007))

8 The complete set of variance decomposition graphs is available upon request.


and empirical perspective (Stock and Watson (2002), Krane (2002), Ahmed, Levin,

and Wilson (2004), Herrera and Pesavento (2005) or Ramey and Vine (2006)).

There is more disagreement instead on the role of improved monetary policy in

the Great Moderation, with some authors advocating its signi�cant role (Clarida,

Gali, and Gertler (2000), Bernanke (2004), Benati (2004) or Boivin and Giannoni

(2006)), and others presenting evidence against this explanation (see, for instance,

Stock and Watson (2002), Stock and Watson (2003), Ahmed, Levin, and Wilson

(2004) or Sims and Zha (2006)). We defer a careful examination of how changes in

the systematic conduct of monetary policy may have a¤ected volatility until section

8.

Instead, the starting point for the analysis of the Great Moderation undertaken

in this section is the robust �nding of Stock and Watson (2002, 2003), who conclude

that �this reduction in volatility is associated with an increase in the precision of

forecasts of output growth� (Stock and Watson (2002), p. 42). Notice that our

framework is a natural candidate to understand the structural causes of this reduc-

tion in forecast errors. In fact, given that our methodology allows for time varying

volatilities and is based on a fully-�edged model, it provides an interpretation for

the structural disturbances hitting the economy.

Figure 5 plots the volatility of GDP growth implied by our model. There are at

least two things to notice from the comparison between �gure 5 and �gure 8, which

reports a simple 10-year moving window estimate of the standard deviation of GDP

growth. First, the DSGE model somewhat overpredicts the level of the volatility

of GDP growth during the entire postwar period. The overprediction would be

far less severe if �gure 8 had been produced using a shorter window, but would

never completely disappear. This problem is common to the time invariant version

of the model and is therefore indicative of di¢ culties in simultaneously matching

the levels of persistence, comovements and volatilities observed in the data, even

with state of the art DSGE models (Del Negro, Schorfheide, Smets, and Wouters

(2006)). Second, nonetheless, the model captures remarkably well the timing and

the size of the Great Moderation, despite the abrupt nature of this fall in volatility.

Observe that the volatility of GDP growth starts declining around 1981, which is

slightly earlier than some estimates provided by the literature using models with

discrete structural breaks. This is due to the speci�cation of our time varying

volatility model, which tends to smooth out abrupt changes (see, for example,


Boivin (2001)). Therefore, we will later compare the �t of our baseline model with

one that allows for a jump in the variance of shocks.

To assess the role played by each shock in accounting for the Great Moderation,

we rely on counterfactual simulations exercises. Our approach consists of using

our stochastic volatility model to simulate the variability of GDP growth under

alternative paths for the standard deviation of each structural disturbance. This

counterfactual simulations can be interpreted as the hypothetical pattern of the

volatility of GDP growth in the period 1981-2004, had the standard deviation of

that particular structural shock only remained unchanged with respect to its 1980

level.9

Figure 6 presents the results of our counterfactual exercises. Our approach gives

a very strong conclusion about the causes of the Great Moderation. As evident from

�gure 6d, the main explanation for the Great Moderation seems to be the sharp

reduction in the volatility of the investment shocks. That is, had the volatility of

these disturbances remained at its 1980 level, then the standard deviation of GDP

growth would have been substantially higher than the one observed in the 1981-2004

period. Finally, it is worth noting that changes in the volatility of the monetary

policy shock had a signi�cant, although much smaller, e¤ect on the decline in the

variance of output growth (�gure 6a).

Even if our main focus is on the volatility of GDP, we perform a similar coun-

terfactual experiment also for the volatility of in�ation, which is also characterized

by a substantial decline around the early 1980s. Similarly to output, the main

contributor to the lower variability of in�ation is the investment shock (�gure 7d).

We will analyze the ability of alternative speci�cations to capture this simultaneous

decline in in�ation and output volatility in section 8.

7. Interpretation of the Main Result

Our results suggest that the key to understand the Great Moderation is to an-

alyze the reasons of the decline in volatility of the investment shocks. Broadly

speaking, these are innovations speci�c to the return on capital or the marginal

e¢ ciency of the investment technology. In this section, we suggest two interpreta-

tions for these shocks and rely on evidence outside our DSGE model to argue for

the plausibility of both views.

9 More precisely, we �x the volatility of each shock to the average of the time varying standarddeviation for all four quarters in 1980. A longer window does not a¤ect our results.


7.1. Investments speci�c technology shocks and the relative price of in-

vestment. A strict, model-based interpretation indicates that the investment shocks

correspond either to investment speci�c technological disturbances or, equivalently,

to shocks to the relative price of investment in terms of consumption goods. In

recent years, data on the latter have been used by other authors to proxy for in-

vestment speci�c technology shocks (see, for instance, Greenwood, Hercowitz, and

Krusell (1997 and 2000) and Fisher (2006)). However, this relative price is not used

in our estimation as we also wish to consider other possible interpretations of the

investment shock. Nonetheless, we �rst verify that the decline in the volatility of

our investment shocks is at least broadly in line with a decline in the variance of

the relative price of investment in terms of consumption goods.

In particular, we construct this relative price using the chain-weighted de�ators

for our components of consumption (non-durables and services) and investment

(durables and total private investment) and estimate the standard deviation of its

growth rate using a simple 10-year moving window. Figure 8 plots the estimate

of this time varying standard deviation and makes clear that the volatility of this

relative price has sharply decreased in the second part of the postwar sample. Not

only does the the timing of this decline coincide remarkably well with the onset

of the Great Moderation, but the patters in the volatilities of the relative price of

investment and GDP growth (also reported in �gure 8) are very similar throughout

the entire postwar period. It is important to stress, however, that the relative price

of investment to consumption has a strong downward trend in the data, which

cannot be accounted for by our model, as we assume that the investment shock is

stationary.10 Nevertheless, we regard the fact that our model provides a very similar

insight (without using any data on the relative price of investment) as a remarkable

result.

A possible criticism to simply looking at the standard deviation of the relative

price of investment is that its decline could have come either from a reduction in the

10 This assumption allows to directly compare our results to a large literature on Bayesianestimation of DSGE models (see, for instance, Del Negro, Schorfheide, Smets, and Wouters (2006),Levin, Onatski, Williams, and Williams (2005) or Smets and Wouters (2006)). Assuming insteada non-stationary investment shock and using data on the relative price of investment in theestimation would require at least two important modi�cations: �rst, we would need to de�ateall real variables by the consumption de�ator (see Greenwood, Hercowitz, and Krusell (1997)or Fisher (2006)), which would then become our measure of price in�ation, providing perhapsa less accurate characterization of monetary policy in the Taylor rule; second, we would needsome additional structural shocks or measurement errors for the observable variables (to avoidstochastic singularity). We leave this extension for future work.


volatility of factors a¤ecting the demand for investment goods (such as monetary

policy) or from a lower volatility of factors a¤ecting the supply of investment goods

(as investment speci�c technological change). Supporting evidence for the latter is

provided by Fisher (2006), who uses a structural vector autoregression (SVAR) to

identify the investment speci�c technology shock as the only disturbance a¤ecting

the relative price of investment in the long run. Fisher (2006) estimates the SVAR in

two separate subsamples (from 1955:I to 1979:II and from 1982:III to 2000:IV) and

�nds that the standard deviation of the identi�ed investment speci�c technology

shock in the second subsample is approximately 70 percent lower. This suggests

that interpreting the lower variability of investment shocks as re�ecting a decline in

the volatility of disturbances to investment speci�c technology would be consistent

with the data.

7.2. Financial frictions. The measure of the relative price of investment analyzed

above does not include �nancial costs related to the purchase of durable and capital

goods. If external �nancing is an important determinant of purchases of this type of

goods, the �nancial cost of borrowing contributes to the e¤ective cost of investment

in terms of consumption goods, which could be re�ected in our �t shock.11 If we

subscribe to this broader interpretation of this disturbance �which is admittedly

outside our current DSGE model�a natural explanation of the Great Moderation

would also be based on a reduction in �nancial frictions.

Interestingly, this purely �theoretical�hypothesis squares remarkably well with

the empirical and anecdotal evidence about the expanded access to credit and bor-

rowing for �rms and households since the beginning of the 1980s. Important ele-

ments of this transformation were the passing of the Depository Institutions Dereg-

ulation and Monetary Control Act (DIDMCA) in 1980, particularly the demise of

regulation Q, and the Garn-St Germain Act of 1982 (see, for instance, Hender-

shott (1990), Dynan, Elmendorf, and Sichel (2005) or Campbell and Hercowitz

(2006)). These changes allowed households unprecedented access to external �-

nancing (Campbell and Hercowitz (2006)), which was further facilitated by the

emergence of secondary mortgage markets (Peek and Wilcox (2006), McCarty and

11 The intuitive link between our shock to the relative price of investment (�t) and �nancialfrictions is even more evident in models that take into consideration agency costs for the �nancingof investment, such as Carlstrom and Fuerst (1997). In particular, Bernanke, Gertler, and Gilchrist(1999) and, more recently, Chari, Kehoe, and McGrattan (2006) have argued that unmodeled�nancial frictions of that kind might be captured in reduced form by disturbances to the realreturn on capital similar to the one we emphasize in this paper.


Peach (2002)). Moreover, �rms�access to external �nancing was enhanced by the

development of a market for bonds with below-investment grade ratings (Gertler

and Lown (1999)), as well as a decline in the cost of new equity issuances (Jermann

and Quadrini (2006)).

Notice that this broader interpretation seems capable of addressing some addi-

tional salient features of the Great Moderation. In particular, as noted by Stock and

Watson (2002), Dynan, Elmendorf, and Sichel (2005) and Justiniano and Primiceri

(2005), the most drastic reduction in volatility has characterized the time series of

durable goods, investment and, especially residential investment. In addition, it is

noteworthy that these events coincided with a decline in credit spreads for mort-

gages and BAA-grade �rms, which is suggestive of a decline in �nancial frictions.12

We want to stress that our model accounts for the link between �nancial frictions

and decreased macroeconomic volatility at best in reduced form and therefore lacks

the structure to understand the underlying transmission mechanisms. In addition,

incorporating �nancial frictions in our model is likely to alter not only the invest-

ment margin that we emphasize in this paper, but other dimensions of the model

as well. This said, our results suggest that further structural analysis of the Great

Moderation would probably bene�t from incorporating an explicit role for these

frictions.

8. Model Fit and Robustness Issues: The Role of Changes in

Monetary Policy and Private Sector Behavior

This section has two objectives. First, we evaluate the �t of our stochastic volatil-

ity model relative to a number of alternative speci�cations. Second, we analyze

changes in the conduct of systematic monetary policy and shifts in the private sec-

tor behavior as potential alternative explanations of the Great Moderation. On this

last point, our counterfactual exercises indicate that, in the context of this model,

the lower variability of U.S. output is di¢ cult to explain when only considering

these changes in parameters and absent shifts in the variance of the disturbances.

As for the assessment of �t, we �nd that our baseline stochastic volatility model

12 For the mortgage market, for instance, the large and volatile spreads of the early 1980scorrespond to the end of a market dominated by heavily regulated thrift institutions, in whichcredit availability was subject to large swings due to �uctuations in deposits (Bradley, Gabriel,and Wohar (1995)). The transition to smaller spreads is commonly associated with the beginningof more e¢ cient and integrated �nancial markets (McCarty and Peach (2002), Hendershott (1990),Dynan, Elmendorf, and Sichel (2005) or Schnure (2005)).


outperforms all alternative speci�cations, even when we allow for a single shift in

the variance of the shocks.

8.1. Model �t. We assess the �t of our model using the marginal likelihood (or

marginal data density), which corresponds to the posterior density integrated over

the model�s parameters. From a Bayesian perspective, the marginal likelihood is

the most comprehensive and accurate measure of �t, as it can be used to construct

posterior odds on competing models.13 The �rst two rows of table 3 report the log-

marginal data density for our model with stochastic volatility and its time invariant

counterpart. As evident, the values of the log-marginal likelihood are strongly in

favor of our stochastic volatility model.

Although the data seem to beg for time variation in the variance of the shocks,

a natural question is whether accounting for a single break in these variances is

enough to capture this salient feature of the data. The answer to this question

is no. The third row of table 3 reports the value of the marginal likelihood for a

model with a single break in the volatilities (set in the �rst quarter of 1984), which

delivers a substantially lower �t than our stochastic volatility baseline.

Rows 4 and 5 of table 3 refer to two alternative speci�cations, in which all coef-

�cients are allowed to change in 1984. The next subsection discusses in detail these

model variants, while here we simply stress that one of them (row 5) incorporates

the possibility that monetary policy may have shifted from passive to active over

the sample.14 Both models deliver a rather poor �t of the data, not only compared

with our baseline model, but also with the model in which only variances shift (row

3). It is also noteworthy that the speci�cation where policy remains always active

outperforms the one in which monetary policy shifts from passive to active.

8.2. The role of changes in monetary policy and private sector behavior.

Two recent studies by Benati and Surico (2006) and Lubik and Surico (2006) have

questioned the usefulness of reduced form methods to analyze the role of monetary

policy in the Great Moderation. These authors recommend, instead, to investigate

this question in the context of structural models. Therefore, despite the poor �t

of both models with time variation in the systematic part of monetary policy and

13 Technical details about the computation of the marginal likelihood are presented in appendixA.

14 As will become clear in the next subsection, monetary policy is denoted as passive whenthe reaction to in�ation is weak and it is not consistent with a detrminate equilibrium.


private sector behavior, we are nonetheless interested in using our DSGE model to

address the possibility that these changes may account for the Great Moderation.

We �rst discuss the estimation of the model with a one time change in all co-

e¢ cients. We then turn to counterfactual experiments that compare the decline

in the variance of GDP growth and in�ation observed in the data with those pre-

dicted by the model under alternative scenarios for the changes in coe¢ cients across

subsamples.

8.2.1. Split sample estimation. We focus on the model in which all coe¢ cients are

allowed to shift across two subsamples: 1953:I-1983:IV and 1984:I-2004:IV.15 We

estimate two variants of this model, one in which values of the policy reaction to

in�ation (��) are always consistent with a unique equilibrium, and an alternative

model where �� is instead consistent with multiple equilibria in the �rst subsample.

Several authors have suggested a shift from passive to active monetary policy as a

prominent explanation of the Great Moderation. Therefore, and since at least in

principle the indeterminacy channel can explain the higher variance of GDP growth

in the pre-Volcker period, we focus mainly on the model with indeterminacy.

In order to construct the likelihood function, we apply the methodology of Lubik

and Schorfheide (2003), who show how to parameterize the continuum of possible

solutions under indeterminacy.16 In this case, the model solution di¤ers in two im-

portant ways from the standard solution under determinacy. First, the transmission

mechanism of the fundamental shocks is not pinned down uniquely by the structural

coe¢ cients and additional free parameters are needed to characterize the solution;

second, sunspot shocks can contribute to the volatility of the model�s endogenous

variables. The reader is referred to appendix C for additional details about the

prior and the model solution in the indeterminate case. Here we point out that,

according to our prior, the direct e¤ect of the sunspot shocks combined with these

changes in the transmission mechanism of the fundamental disturbances results in

substantially higher a-priori model implied standard deviations for all series relative

to our benchmark determinate model. Moreover, we only consider regions of the

parameter space in which the degree of indeterminacy is one and indeterminacy

15 We split the sample in 1984 because it corresponds to the onset of the Great Moderation.Justiniano and Primiceri (2005) perform a similar exercise excluding the 1979 - 1983 period andreach very similar conclusions.

16 When estimating the indeterminate model, we modify our prior for �� for the �rst subsamplefollowing Lubik and Schorfheide (2004), and specify a Gamma distribution with mean equalto 1:1 and standard deviation equal to 0:5. This prior assigns roughly equal probability to thedeterminacy and indeterminacy regions of the parameter space determined by �� .


is generated by low values of ��, i.e. low interest rate reactions to in�ation. This

means that we e¤ectively truncate our prior at the boundary of a multi-dimensional

indeterminacy region.17

Table 4 presents medians, 5th and 95th percentiles of the posterior distribution of

the coe¢ cients estimated over the two subsamples under indeterminacy before 1984.

Consistent with the results of the stochastic volatility model, there are important

changes in the standard deviation of the shocks across sample periods. However,

some of the remaining coe¢ cient estimates exhibit di¤erences across subsamples as

well. By construction, this is particularly evident for the policy reaction to in�ation,

��, which displays a considerable increase, switching from a value implying multiple

equilibria to a value consistent with a unique equilibrium (Clarida, Gali, and Gertler

(2000)). For space considerations we do not report estimates for the model in which

policy is always active but simply note that coe¢ cient changes are less pronounced

in this case.

8.2.2. Counterfactual experiments. Our aim is to understand whether changes in

monetary policy and private sector coe¢ cients across subsamples can induce a

reduction in the volatility of the endogenous variables similar to the one observed

in the data. Once again, we focus on the results of the indeterminate model and

brie�y return to the determinate case towards the end.

As a starting point, the �rst column of table 5a shows that the unconditional

standard deviation of GDP growth in the second sample period relative to the

�rst is 0:76, whilst this ratio is 0:26 for in�ation.18 In the context of our model

this decline can potentially be explained by three di¤erent sets of parameters: the

standard deviations of the shocks, monetary policy coe¢ cients and the remaining

structural coe¢ cients. Our goal here is to isolate the contribution of the last two.

We begin by assessing the role of a policy shift from passive to active on the

variability of output growth and in�ation. To this end, we compare the model

implied standard deviation of these two variables when the coe¢ cients of the Taylor

17 In this model it is not possible to characterize the region of indeterminacy analytically.While the strength of the policy reaction to in�ation is the crucial parameter, extensive simulationanalysis suggests that determinacy of equilibrium also hinghes on the remaining coe¢ cients of theTaylor rule. Moreover, indeterminacy also seems to occur for very large values of the degree ofwage regidity, which are substantially further out from the right tail of our posterior estimates.As mentioned above, in our exercise we divide the determinacy and indeterminacy regions of theparameter space based only on the reponse of in�ation in the Taylor rule.

18 Notice that, compared to the data, the model of this section underpredicts the decline involatility of GDP growth and overpredicts the decline in volatility of in�ation. This is also relatedto the poor �t of this model, documented in section 8.1.


rule estimated in the second subsample replace those of the �rst subsample, leaving

all other coe¢ cients unchanged. These counterfactual standard deviations are given

in the second column of table 5a (relative to the volatility estimates in the �rst

subsample). Similarly, the third column of table 5a corresponds to the relative

standard deviations when we redo this counterfactual exercise but replacing, in

addition, the estimated coe¢ cients of the private sector from the second sample

period. Finally, the last column presents the ratio of standard deviations once we

also replace the autocorrelation of the shocks from the second subsample.

Table 5a makes some important points. First, if monetary policy in the 1960s

and 1970s had been as aggressive against in�ation and as weak in response to real

activity as in the later part of the sample, we would have probably experienced

substantially lower in�ation volatility. However, the variability of GDP growth

would have been, if anything, even higher.19 This result is in line with Casteln-

uovo (2006) and Boivin and Giannoni (2006) who also note that more aggressive

monetary policy would have increased output variability, absent a change in the

shocks� process. One explanation for this result is that the model attributes a

large share of output variability to �supply�shocks �i.e. the investment shocks and

neutral technology disturbances�which in the �rst subsample imply a strong neg-

ative correlation between in�ation and output. Hence, counterfactually imposing a

policy response that is more accomodating towards output and substantially more

aggressive towards in�ation leads, all else equal, to an increase in the variability of

output.

Our second counterfactual indicates that jointly considering changes in mone-

tary policy and the private sector behavior could have reduced the variability of

in�ation but, once again, not output growth. Our last counterfactual suggests that

changes in the autocorrelation of the disturbances could have in�uenced the implied

standard deviations as well. These changes, however, would have induced a higher

volatility of output and all other variables (absent a decline in the variance of the

shocks), which is at odds with the data. On this last point, we note that this is

mainly related to a substantial increase in the autocorrelation of the labor disutility

19 More precisely, the volatility of the output gap is substantial in our model, especially inthe �rst subsample. The stronger response to in�ation and weaker response to real activity wouldhave somewhat lowered this volatlity. However, the counterfactual policy would have also alteredthe correlation between the output gap and potential output in such a way that, as a results, wewould have observed more volatile output.


shock in the post-84 sample, which is almost fully compensated by an even more

sizable reduction in its volatility.20

Table 5b performs similar counterfactual simulations for the split-sample model

in which policy is active throughout the entire sample. In this case �� is estimated

at 1.63 for the �rst subsample, hence the policy shift in 1984 is less dramatic than

in the indeterminate case. Accordingly, counterfactual simulations exhibit milder

variations in response to the policy change. Variations in autocorrelation coe¢ cients

are more subdued in this case as well, which is re�ected in a less drastic change in

variances in the last column. However, the general picture from this counterfactual

exercise is broadly in line with the results of table 5a.

Summarizing, changes in the systematic conduct of monetary policy may have

contributed to the reduction in in�ation variability. However, table 5 makes evident

that neither changes in monetary policy nor the remaining model coe¢ cients -other

than a fall in the volatilities- can account for the observed simultaneous decline

in the variability of output and in�ation. This is true even when we allow for

indeterminacy and multiple equilibria in the pre-1984 period, although the data

would seem to favor the determinate case.

On balance, however, we mention some possible caveats to the results of this

section. In particular, allowing for all coe¢ cients to change may be prone to identi-

�cation issues in a large scale model like ours estimated on relatively short subsam-

ples. This is perhaps more so for the model with passive policy, if the identi�cation

of indeterminacy is rather weak (Beyer and Farmer (2006)). While the risk of

over�tting is signi�cant for both models, this seems particularly the case for the

indeterminate model, as re�ected in the marginal data density reported in table 3.21

Finally, one possible explanation for the poor �t of all models with policy shifts

is the fact that we have forced the change in policy to occur at the time of the

Great Moderation (i.e. in 1984). In this respect, a promising direction for future

work would be to combine the time varying volatility model of this paper with the

20 Indeed, simulations not reported in table 5 indicate that the two e¤ects counteract each otheralmost entirely. These large changes in the persistence and volatility of the labor disutility shockmight are probably due to weak identi�cation in the short subsamples. In fact, these changes aresmaller under determinacy since it is well known that indeterminacy generates higher persistence.

21 To account for problems of over�tting, table 3 (rows 6 and 7) also reports the log-marginallikelihood for two additional versions of the model, one in which only monetary policy and volatili-ties undergo a one time change in 1984, and another one in which only monetary policy can switch(in both cases from passive to active) at the same date. The �t of these two models is �1941:1and �1983:4 respectively, which, in both cases is lower than the �t of the model with a single shiftin variances.


time varying coe¢ cients speci�cation of Fernandez-Villaverde and Rubio-Ramirez

(2007).

The fact that we do not �nd a big role for improved monetary policy in the

decline of output volatility is consistent with evidence provided by Stock and Wat-

son (2003), Ahmed, Levin, and Wilson (2004), Primiceri (2005) and Sims and

Zha (2006). Di¤erently from these authors, however, our analysis is conditional

on a DSGE model, with all the pros and cons that this entails. One advantage

of a structural approach is that counterfactual exercises involving changes in pol-

icy are easier to interpret. Of course, this comes at the cost of imposing a set

of stronger identifying restrictions on the analysis. Therefore, we cannot exclude

the possibility that there are other less standard models in which an explanation

for the Great Moderation based on changes in monetary policy may fare better.

For example, we suspect that deviating from full information rational expectations

might be promising in this direction (see, for instance, Walsh (2005), Orphanides

and Williams (2005), Erceg and Levin (2003)), although we do not entertain this

possibility in this paper.

9. Concluding Remarks

In this paper we have estimated a large scale DSGE model of the U.S. business

cycle allowing for the volatility of the structural disturbances to change over time.

Our results indicate that the volatility of several structural shocks has changed

dramatically in the postwar period. However, the sharp reduction in the standard

deviation of GDP growth that has characterized the last twenty years can be ex-

plained mostly due to the decline in the variability of a single shock: the shock

speci�c to the equilibrium condition of investment.

We have suggested two interpretations of this �nding. The �rst view is strictly

model-based and suggests that these disturbances correspond either to investment

speci�c technological shocks or, equivalently, to innovations in the relative price

of investment in terms of consumption goods. In addition, since a broad interpre-

tation of this relative price should include the �nancial costs of borrowing for the

purchase of investment goods, we have also suggested an interpretation of the Great

Moderation based on a decline in �nancial frictions.

More generally, our results point to dramatic changes in the investment equilib-

rium condition that played a prominent role in the Great Moderation.


Appendix A. The Estimation Algorithm

A.1. The standard case: homoskedastic disturbances. For the model with-

out stochastic volatility, the estimation algorithm is a random walk Metropolis

MCMC procedure, as suggested originally by Schorfheide (2000). We �rst use a

maximization algorithm (Chris Sims�csminwel) to �nd the posterior mode and to

obtain an inverse Hessian, which then becomes the dispersion measure for our pro-

posal distribution. This is done for multiple initial values (at least 30) drawn at

random from our prior to ensure convergence of this initial search to a unique mode.

We then scale this variance-covariance matrix to attain an acceptance rate close to

0:25, as it is usually suggested. Appendix B discusses convergence diagnostics which

we also apply to this posterior simulator.

A.2. Stochastic volatility. When the structural shocks exhibit stochastic volatil-

ity, this algorithm must be modi�ed to account for inference on the unobserved

stochastic volatilities. A Metropolis within Gibbs MCMC algorithm allows us to

iteratively draw from the posterior densities of the DSGE model�s parameters, sto-

chastic volatilities and associated innovation variances. Some steps of the algorithm

require data augmentation (Tanner and Wong (1987)) using the Gibbs sampler in

order to draw other latent auxiliary variables (Chib (2001)). For instance, as dis-

cussed below, generating a draw for the stochastic volatilities entails using a normal

mixture approximation and sampling a set of latent indicators for the components

of this mixture.

To illustrate the steps involved in sampling from the di¤erent blocks, let the

vector � collect all parameters of the DSGE model (other than the standard devi-

ations of the structural disturbances of the time invariant model) and notice that

the solution of the log-linearized DSGE model leads to a state-space representation

of the form

xt = D�t(A.1)

�t = A(�)�t�1 +B(�)�t(A.2)

where xt represents a vector of observable variables and �t denotes the vector of

endogenous / state variables in log-deviation from the deterministic steady state.

As discussed in section 2 the novelty of our framework is that the vector of struc-

tural innovations �t (dimension n � 1) is allowed to have a time varying variance


covariance matrix. Indexing each structural shock by i, the stochastic volatilities

for each shocks are modelled as

�i;t = �i;t"i;t(A.3)

log �i;t = (1� ��i) log �i + ��i log �i;t�1 + �i;t(A.4)

"i;t � N(0; 1)(A.5)

�i;t � N(0; !2i ) i = 1; :::; n.

Let the vector ht, with entry i given by hi;t = log �i;t, collect the log volatilities

for all shocks at time t and stack the whole sample of stochastic volatilities into

the matrix HT � [h1; h2; :::; ht; :::; hT ]0. Finally, we denote the sample of structuralshocks as �T � [�1;�2; :::; �t; :::; �T ]0 and the vector including all the �xed coe¢ cientsof the volatility processes by � �

��1; ��1 ; !

21; :::; �n; ��n ; !

2n

�.

Suppose that the MCMC algorithm has completed iteration g (> 0); producing

samples �(g); HT;(g) and �(g):of the parameters of interest (individual elements of

a vector are indexed by i while (g) indicates the current state of the chain). In

iteration g + 1, the following �ve steps are used to generate a set of new draws.

A.2.1. Step 1: Draw the structural shocks �T;(g+1). In order to generate a new sam-

ple of the stochastic volatilities we must �rst obtain a new draw of the structural

shocks. This can be done easily using the e¢ cient simulation smoother for dis-

turbances developed by Durbin and Koopman (2002). The simulation smoother is

applied to the state space representation given by (A.1) and (A.2).

A.2.2. Step 3: Draw the stochastic volatilities HT;(g+1). With a draw of �T

in hand the system of nonlinear measurement equations in (A.3) for each structural

shock, can be easily converted in a linear one, by squaring and taking logarithms

of every element. Due to the fact that the squared shocks �2i;t can be very small,

an o¤set constant is used to make the estimation procedure more robust. Dropping

the iteration indicators momentarily for ease of notation, this leads to the following

approximating state space form:

~�i;t = 2hi;t + ei;t(A.6)

hi;t = hi;t�1 + �i;t.(A.7)

where ~�i;t = log[(�i;t)2 + �c]; �c is the o¤set constant (set to 0:001); ei;t = log("2i;t).

Observe that the e�s and the ��s are not correlated. The resulting system has a


linear, but non-Gaussian state space form, because the innovations in the measure-

ment equations are distributed as a log�2(1). In order to further transform the

system in a Gaussian one, a mixture of normals approximation of the log�2(1)

distribution is used, as described in Kim, Shephard, and Chib (1998). Under the

assumption of orthogonality across the "�s (recall the variance covariance matrix

of the "�s is the identity matrix) this implies that the variance covariance matrix

of the v�s is also diagonal, which justi�es using the same (independent) mixture of

normals approximation for each innovation:

f(ei;t) =KXk=1

qkfN (ei;tjsi;t = k), i = 1; :::; n

where si;t is the indicator variable selecting which member of the mixture of nor-

mals has to be used at time t for the innovation i, qk = Pr(si;t = k) and fN (�)denotes the pdf of a normal distribution. Kim, Shephard, and Chib (1998) select

a mixture of 7 normal densities (K = 7) with component probabilities qk, means

mk � 1:2704, and variances r2k, j = 1; :::; 7, chosen to match a number of momentsof the log�2(1) distribution. For completeness the constants are reported below

fqj ;mj ; r2jg below.22

! qj = Pr(! = j) mj r2j

1 0.00730 -10.12999 5.79596

2 0.10556 -3.97281 2.61369

3 0.00002 -8.56686 5.17950

4 0.04395 2.77786 0.16735

5 0.34001 0.61942 0.64009

6 0.24566 1.79518 0.34023

7 0.25750 -1.08819 1.26261

Source: Kim, Shephard and Chib (1998).

Conditional on sT;(g), the system has an approximate linear and Gaussian state

space form. Therefore a new draw for the complete history of the volatilityHT;(g+1)can

be obtained recursively with the standard Gibbs sampling for state space forms us-

ing, for instance, the forward-backward recursion of Carter and Kohn (1994).

22 We abstract from the reweighting procedure used in Kim, Shephard, and Chib (1998) tocorrect the minor approximation error.


A.2.3. Step 3: Draw the indicators of the mixture approximation sT;(g+1). A new

sample of the indicators, s(g+1)i;t , for the mixture is obtained conditional on �T;(g+1)

and HT;(g+1) by independently sampling each from the discrete density de�ned by

Pr(s(g+1)i;t = j j ~�(g+1)i;t ; h

(g+1)i;t ) _ qjfN (~�(g+1)i;t j2h(g+1)i;t +mj�1:2704; r2j ), j = 1; :::; 7

Consistent with notation above, collect the indicators for which component of the

mixture of the normal approximation to use for each structural shock and time

period into a stacked matrix sT;(g+1) = [s(g+1)1 ; s(g+1)2 ; :::; s

(g+1)t ; :::; s

(g+1)T ]0

A.2.4. Step 4: Draw the coe¢ cients of the stochastic volatility processes.

Having generated a sample HT;(g+1), the elements of the vector �(g+1) can be

generated easily from usual Normal inverse-Gamma distributions.

A.2.5. Step 5: Draw the DSGE parameters �(g+1). As in the time invariant al-

gorithm, a new candidate parameter �� is drawn from a proposal density. However,

in this case, the computation of the likelihood used to construct the probability

of acceptance depends on HT;(g+1). More formally the candidate draw is accepted

with probability

a = min

(1;

L(Xj��;HT;(g+1))�(��)

L(Xj�(g) ;HT;(g+1))�(�(g))

),

where X is the matrix of data and L(�) and �(�) denote the likelihood and the priordistribution respectively.

These �ve steps are repeated N times, across multiple chains.

We conclude by observing that the computational demands of this algorithm are

substantial. As a benchmark, generating and storing 100 draws takes roughly 11

minutes in a computer using an AMD Athlon 64 processor 3000+ at 1.81 Ghz with

1 MB of RAM.

A.3. Computation of the marginal likelihood. As it is standard in the litera-

ture, for the computation of the marginal likelihood of the time invariant model we

use the modi�ed harmonic mean method of Gelfand and Dey (1994) and Geweke

(1998).

For the computation of the marginal likelihood of the model with stochastic

volatility we also use a version of the modi�ed harmonic mean method. In fact, it

is easy to show that

m(X) =

�Zf(�;HT )

L(Xj�;HT ) � � (�;HT )� p(�;HT jX) � d(�;HT )

��1,


where X is the matrix of data, m(X) denotes the marginal data density and L (�),�(�) and p(�) denote the sampling, the prior and the normalized posterior densitiesrespectively. f (�) can be any pdf with support contained in the support of theposterior density. For computational convenience, we choose f(�;HT ) = f (�) �f(HT ). Moreover, following Geweke (1998), f (�) is chosen to be a truncated

multivariate normal with mean and variance equal to the mean and the variance of

the posterior draws of �. Since the dimension of HT is very large we have decided to

set f(HT ) = �(HT ). It follows that the marginal likelihood can be approximated

by

�mN (X) =

24 1N

NXj=1

f(�j)

p(Xj�j ;HTj ) � � (�j)

35�1 ,where �j and HT

j are draws from the posterior distribution and we have used the

the fact that our priors for � and HT are independent.

�m(X) is very easy to compute and converges (almost surely) to m(X), such

that it represents a consistent estimate of marginal likelihood (Newton and Raftery

(1994)). However, it might not be stable as it may not satisfy a Gaussian central

limit theorem because the random variable f(�j)

p(Xj�j ;HTj )��(�j)

can violate the assump-

tion of a �nite variance. Nevertheless, in the particular application carried out in

this paper, we have noted that this method works quite well, delivering estimates of

the marginal likelihood which are almost identical across multiple simulation chains

and models with slightly di¤erent priors.

We have nonetheless also tried to compute the marginal likelihood using a com-

bination of the modi�ed harmonic mean and the method of Chib (1995). Our

experience indicates that this alternative method is very unreliable and extremely

slow to converge. Indeed di¤erent chains produced wide variation in estimates of

the marginal likelihood although posterior coe¢ cient and volatility estimates were

almost identical across runs.

Appendix B. Convergence

We assess the convergence of our posterior simulators using a battery of diag-

nostics.

For the stochastic volatility model, we launch multiple chains of our Metropolis

within Gibbs simulator from di¤erent starting values (drawn randomly from the

prior). To check that these agree in their characterization of the posterior distribu-

tion we look at various sample moments within and across chains. Several chains of


di¤erent length initialized in this manner delivered roughly identical results when

looking at means, medians, posterior percentiles as well as trace and kernel plots.

More formally, for the multiple chains used to generate the results in the paper,

table 6 reports potential scale reduction factors proposed by Brooks and Gelman

(1998) both for variances and 95 percent intervals (�rst two columns). These num-

bers are very close to one and therefore well below the 1.2 benchmark which is

widely used in practice as an upper bound for convergence. In addition, the last

two columns test for the equality of means using Geweke�s (1992) estimator on the

initial 20 and last 50 percent of the sample, using two alternative estimators for

the serially correlated variance of the draws. At the 95 percent con�dence level we

cannot reject the null hypothesis of equal means for all but one coe¢ cient. For this

single case, trace plots suggest that this is due to a few outliers in one of the chains,

while means and medians across chains di¤er by less than 0.01.

Appendix C. The Solution Method under Indeterminacy

In order to solve the model we log-linearize (2.1) and obtain

~�0(�)yt = ~�1(�)yt�1 + ~�2(�)Etyt+1 + ~�3(�)�t,

where the �hat� denotes log deviations from the non-stochastic steady state and~�0, ~�1, ~�2 and ~�3 are matrices conformable to yt and �t. We de�ne a vector of

endogenous forecast errors (!t) and extend the vector of endogenous variables to

include the expectational variables. Denoting this extended vector by �t, we can

put the system in Sims (2001) canonical form:

�0(�)�t = �1(�)�t�1 +(�)�t +�(�)!t.

We can now solve for the endogenous forecast error with the methodology of Lubik

and Schorfheide (2003) and Lubik and Schorfheide (2004) and write the solution of

the model as

�t = A(�)�t�1 +B(�)�t| {z }part I

+ C(�)�~M�t + st

�| {z }

part II

.

Part I of this expression corresponds to the solution of the model under determinacy,

which is completely pinned down by the structural coe¢ cients of the model (�).

Part II of the expression is the additional term that appears and parameterizes the

continuum of equilibria under indeterminacy. Note that it includes a modi�cation

of the transmission mechanism of the fundamental shocks (parameterized by ~M)


as well as the e¤ect of a sunspot shock (st � N�0; �2s

�). Details regarding the

derivation of the solution and the interpretation of the matrices can be found in

Lubik and Schorfheide (2003) and Lubik and Schorfheide (2004).

In the indeterminacy region, we only consider draws for which the degree of

indeterminacy is one and discard all other draws. In this case, the dimensions of~M and �s are n� 1 and 1� 1 respectively.An important issue is of course the speci�cation of a prior density for these ad-

ditional free parameters. For �s, we choose an Inverse-Gamma prior with mean

and standard deviation equal to 0:15. For ~M , we follow Lubik and Schorfheide

(2004) and center our prior on M�(�), where M�(�) denotes the continuity solu-

tion, i.e. the value of ~M for which the impact of fundamental shocks on endogenous

variables is continuous at the boundary of the indeterminacy region. For a given

parameter value � leading to indeterminacy, M�(�) is found by minimizing (using

a least squares criterion) the distance between the impact matrix under indeter-

minacy and the one at the boundary of the determinacy region. Because of the

dependence of M� on � it is convenient to center our prior on the continuity so-

lution by instead specifying a zero mean prior (with standard deviation equal to

0:5) for M � ~M �M�(�). Hence, for each indeterminate draw, we must �nd the

corresponding parameter value at the boundary of the determinacy region. We do

this by gradually increasing ��, while keeping all other coe¢ cients �xed until we

obtain a determinate solution. The shape of the determinacy and indeterminacy

regions of our parameter space is slightly more complex than in simpler three equa-

tions New-Keynesian models. Therefore, in very few cases this procedure does not

generate a determinate solution and we discard the draw. For the model in which

all coe¢ cients are allowed to change (table 4), for instance, this entailed discarding

0:13 percent of the draws.

References

Ahmed, S., A. Levin, and B. A. Wilson (2004): �Recent U.S. Macroeconomic Stability: Good

Policies, Good Practicies, or Good Luck?,� The Review of Economics and Statistics, 86(3),

824�832.

Altig, D., L. J. Christiano, M. Eichenbaum, and J. Linde (2005): �Firm-Speci�c Capital,

Nominal Rigidities and the Business Cycle,�NBER Working Paper No. 11034.

An, S., and F. Schorfheide (2006): �Bayesian Analysis of DSGE Models,� Econometric Re-

views, forthcoming.


Benati, L. (2004): �Monetary Rules, Indeterminacy, and the Business-Cycle Stylised Facts,�

European Central Bank, mimeo.

Benati, L., and P. Surico (2006): �VAR Analysis and the Great Moderation,�European Central

Bank, mimeo.

Bernanke, B. (2004): �Great Moderation,�Speech, February 20, 2004.

Bernanke, B., andM. Gertler (1989): �Agency Costs, Net Worth, and Business Fluctuations,�

American Economic Review, 79, 14�31.

Bernanke, B., M. Gertler, and S. Gilchrist (1999): �The Financial Accelerator in a Quan-

titative Business Cycle Framework,� in Handbook of Macroeconomics, ed. by J. B. Taylor, and

M. Woodford. North Holland, Amsterdam.

Bernanke, B. S., and I. Mihov (1998): �Measuring Monetary Policy,�The Quarterly Journal

of Economics, 113, 869�902.

Beyer, A., and R. Farmer (2006): �On the Indeterminacy of Determinacy and Indeterminacy:

Comments on the Paper "Testing for Indeterminacy" by Thomas Lubik and Frank Schorfheide,�

American Economic Review, Forthcoming.

Bils, M., and P. J. Klenow (2004): �Some Evidence on the Importance of Sticky Prices,�

Journal of Political Economy, 112(5), 947�985.

Blanchard, O. J., and J. Simon (2001): �The Long and Large Decline in U.S. Output Volatility,�

Brooking Papers on Economic Activity, 2001-1, 135�174.

Boivin, J. (2001): �The Fed�s Conduct of Monetary Policy: Has It Changed and Does It Matter?,�

mimeo, Columbia University.

Boivin, J., and M. Giannoni (2006): �Has Monetary Policy Become More E¤ective?,�Review

of Economics and Statistics, 88(3), 445�462.

Bradley, M. G., S. A. Gabriel, andM. E. Wohar (1995): �The Thrift Crisis, Mortgage-Credit

Intermediation, and Housing Activity,� Journal of Money, Credit, and Banking, 27(2), 476�97.

Brooks, S. P., and A. Gelman (1998): �General Methods for Monitoring Convergence of Iter-

ative Simulations,� Journal of Computational and Graphical Statistics, 7, 434�455.

Calvo, G. (1983): �Staggered Prices in a Utility-Maximizing Framework,� Journal of Monetary

Economics, 12(3), 383�98.

Campbell, J. R., and Z. Hercowitz (2006): �The Role of Collateralized Household Debt in

Macroeconomic Stabilization,�mimeo, Federal Reserve Bank of Chicago.

Canova, F., L. Gambetti, and E. Pappa (2005): �The Structural Dynamics of US Output and

In�ation: What Explains the Changes?,�mimeo, Bocconi University.

Carlstrom, C. T., and T. S. Fuerst (1997): �Agency Costs, Net Worth, and Business Fluc-

tuations: A Computable General Equilibrium Analysis,� American Economic Review, 87(5),

893�910.

Carter, C. K., and R. Kohn (1994): �On Gibbs Sampling for State Space Models,�Biometrika,

81(3), 541�553.

Castelnuovo, E. (2006): �Assessing Di¤erent Drivers of the Great Moderation in the U.S.,�

University of Padua, mimeo.

Chari, V., P. J. Kehoe, and E. R. McGrattan (2006): �Business Cycle Accounting,�Econo-

metrica, forthcoming.


Chauvet, M., and S. Potter (2001): �Recent Changes in the U.S. Business Cycle,�Manchester

School of Economics and Social Studies, 69(5), 481�508.

Chib, S. (1995): �Marginal Likelihood from the Gibbs Output,�Journal of the American Statis-

tical Association, 90, 1313�1321.

(2001): �Markov Chain Monte Carlo Methods: Computation and Inference,�in Handbook

of Econometrics: Volume, ed. by J. J. Heckman, and E. Leamer, pp. 3569�3649. North Holland,

Amsterdam.

Christiano, L. J., M. Eichenbaum, and C. L. Evans (2005): �Nominal Rigidities and the

Dynamic E¤ect of a Shock to Monetary Policy,� The Journal of Political Economy, 113(1),

1�45.

Clarida, R., J. Gali, and M. Gertler (2000): �Monetary Policy Rules and Macroeconomic

Stability: Evidence and Some Theory,�The Quarterly Journal of Economics, 115(1), 147�180.

Cogley, T., and T. J. Sargent (2005): �Drifts and Volatilities: Monetary Policies and Out-

comes in the Post WWII U.S.,�Review of Economic Dynamics, 8(2), 262�302.

Del Negro, M., F. Schorfheide, F. Smets, and R. Wouters (2006): �On the Fit and Fore-

casting Performance of New Keynesian Models,� Journal of Business and Economic Statistics,

Forthcoming.

Durbin, J., and S. J. Koopman (2002): �A Simple and E¢ cient Simulation Smoother for State

Space Time Series Analysis,�Biometrika, 89(3), 603�616.

Dynan, K. E., D. W. Elmendorf, and D. E. Sichel (2005): �Can Finacial Innovation Explain

the Reduced Volatility of Economic Activity?,� in Carnegie-Rochester Conference Series on

Public Policy, ed. by B. McCallum, and C. I. Plosser.

Erceg, C. J., D. W. Henderson, and A. T. Levin (2000): �Optimal Monetary Policy with

Staggered Wage and Price Contracts,� Journal of Monetary Economics, 46(2), 281�313.

Erceg, C. J., and A. T. Levin (2003): �Imperfect Credibility and In�ation Persistence,�Journal

of Monetary Economics, 50(4), 915�944.

Fernandez-Villaverde, J., and J. Rubio-Ramirez (2006): �Estimating Macroeconomic Mod-

els: A Likelihood Approach,�Review of Economic Studies, forthcoming.

(2007): �How Structural Are Structural Parameters?,�in NBER Macroeconomics Annual.

Fisher, J. D. M. (2006): �The Dynamic E¤ect of Neutral and Investment-Speci�c Technology

Shocks,� Journal of Political Economy, 114(3), 413�451.

Gelfand, A. E., and D. K. Dey (1994): �Bayesian Model Choice: Asymptotics and Exact

Calculations,� Journal of the Royal Statistical Society (Series B), 56(3), 501�514.

Gertler, M., and C. Lown (1999): �The Information in the High-Yield Bond Spread for the

Business Cycle: Evidence and Some Implications,� Oxford Review of Economic Policy, 15(3),

132�150.

Geweke, J. F. (1992): �Evaluating the Accuracy of Sampling-Based Approaches to the Calcu-

lation of Posterior Moments,� in Bayesian Statistics, ed. by J. Berger, J. Bernardo, A. Dawid,

and A. Smith, pp. 169�194. Oxford University Press, Oxford, UK.

(1998): �Using Simulation Methods for Bayesian Econometric Models: Inference, Devel-

opment and Communication,�mimeo, University of Iowa.


Greenwood, J., Z. Hercowitz, and P. Krusell (1997): �Long Run Implications of Investment-

Speci�c Technological Change,�American Economic Review, 87(3), 342�362.

Greenwood, J., Z. Hercowitz, and P. Krusell (2000): �The role of investment-speci�c tech-

nological change in the business cycle,�European Economic Review, 44(1), 91�115.

Hendershott, P. H. (1990): �The Market for Home Mortgage Credit: Recent Changes and

Future Prospects,�NBER Working Paper no. 3548.

Herrera, A. M., and E. Pesavento (2005): �The Decline in U.S. Output Volatility: Structural

Changes and Inventory Investment,� Journal of Business and Economic Statistics, 23(4), 462�

472.

Ireland, P. N. (2004): �Technology Shocks in the New Keynesian Model,�The Review of Eco-

nomics and Statistics, 86(4), 923�936.

Jermann, U., and V. Quadrini (2006): �Financial Innovations and Macroeconomic Volatility,�

University of Southern California, mimeo.

Justiniano, A., and G. E. Primiceri (2005): �The Time Varying Volatility of Macroeconomic

Fluctuations,�NBER Working Paper no. 12022.

Kahn, A., and J. K. Thomas (2007): �Inventories and the Business Cycle: An Equilibrium

Analysis of (S,S) Policies,�American Economic Review, Forthcoming.

Kahn, J. A., and M. M. McConnell (2002): �Has Inventory Volatility Returned? A Look

at the Current Cycle,� Federal Reserve Bank of New York Current Issues in Economics and

Finance, 8(5), 1�6.

Kim, C.-J., and C. R. Nelson (1999): �Has the U.S. Economy Become More Stable? A Bayesian

Approach Based on a Markov-Switching Model of the Business Cycle,�The Review of Economics

and Statistics, 81, 608�616.

Kim, S., N. Shephard, and S. Chib (1998): �Stochastic Volatility: Likelihood Inference and

Comparison with ARCH Models,�The Review of Economic Studies, 65(3), 361�393.

Krane, S. (2002): �Commentary,�FRBNY Economic Policy Review, pp. 203�206.

Laforte, J.-P. (2005): �DSGE Models and Heteroskedasticity: A Markov-Switching Approach,�

mimeo, Board of Governors of the Federal Reserve System.

Levin, A. T., A. Onatski, J. C. Williams, and N. Williams (2005): �Monetary Policy Under

Uncertainty in Micro-Founded Macroeconometric Models,� in NBER Macroeconomics Annual.

Lubik, T. A., and F. Schorfheide (2003): �Computing Sunspot Equilibria in Linear Rational

Expectations Models,� Journal of Economic Dynamics and Control, 28, 273�285.

(2004): �Testing for Indeterminacy: An Application to U.S. Monetary Policy,�American

Economic Review, 94(1), 190�217.

Lubik, T. A., and P. Surico (2006): �The Lucas Critique and the Stability of Empirical Models,�

mimeo, Federal Reserve Bank of Richmond.

Lucca, D. O. (2005): �Resuscitating Time to Build,�mimeo, Northwestern University.

Maccini, L. J., and A. Pagan (2003): �Explaining the Role of Inventories in the Business Cycle,�

mimeo, Johns Hopkins University.

McCarty, J., and R. W. Peach (2002): �Monetary Policy Transmission to Residential Invest-

ment,�Federal Reserve Bank of New York Economic Policy Review, 8(1), 139�158.


McConnell, M. M., and G. Perez-Quiros (2000): �Output Fluctuations in the United States:

What Has Changed Since the Early 1980�s,�American Economic Review, 90(5), 1464�1476.

Newton, M. A., and A. E. Raftery (1994): �Approximate Bayesian Inference with the

Weighted Likelihood Bootstrap,� Journal of the Royal Statistical Society, Series b (Method-

ological), 56(1), 3�48.

Orphanides, A., and J. C. Williams (2005): �The Decline of Activist Stabilization Policy:

Natural Rate Misperceptions, Learning, and Expectations,�Journal of Economic Dynamics and

Control, 29, 1927�1950.

Peek, J., and J. A. Wilcox (2006): �Housing, Credit Constraints, and Macro Stability: The

Secondary Mortgage Market and Reduced Cyclicality of Residential Investment,�American Eco-

nomic Review, 96(2), 135�140.

Primiceri, G. E. (2005): �Time Varying Structural Vector Autoregressions and Monetary Policy,�

The Review of Economic Studies, 72, 821�852.

Ramey, V. A., and D. J. Vine (2006): �Declining Volatility in the U.S. Automobile Industry,�

American Economic Review, 96(5).

Rotemberg, J. J., and M. Woodford (1997): �An Optimization-Based Econometric Model for

the Evaluation of Monetary Policy,�NBER Macroeconomics Annual, 12, 297�346.

Schnure, C. (2005): �Boom and Bust Cycles in Housing: The Changing Role of Financial Struc-

ture,� IMF Working Paper no. 200.

Schorfheide, F. (2000): �Loss-Function Based Evaluation of DSGE Models,�Journal of Applied

Econometrics, 15, 645�670.

Shephard, N. (1994): �Partial Non-Gaussian State Space,�Biometrika, 81, 115�131.

Sims, C. A. (2001): �Solving Linear Rational Expectations Models,� Journal of Computational

Economics, 20(1-2), 1�20.

Sims, C. A., and T. Zha (2006): �Were There Regime Switches in US Monetary Policy?,�

American Economic Review, 96(1), 54�81.

Smets, F., and R. Wouters (2003): �An Estimated Stochastic Dynamic General Equilibrium

Model of the Euro Area,� Journal of the European Economic Association, 1(5), 1123�1175.

(2006): �Shocks and Frictions in US Business Cycles: A Bayesian Approach,�American

Economic Review, forthcoming.

Stock, J. H., and M. W. Watson (2002): �Has the Business Cycle Changed and Why?,� in

NBER Macroeconomic Annual, ed. by M. Gertler, and K. Rogo¤. MIT Press, Cambridge, MA.

(2003): �Has the Business Cycle Changed? Evidence and Explanations,�in FRB Kansas

City Symposium, Jackson Hole, Wyoming.

Tanner, M. A., and W. H. Wong (1987): �The Calculation of Posterior Distributions by Data

Augmentation,� Journal of the American Statistical Association, 82, 528�550.

Walsh, C. E. (2005): �Comments on �Monetary Policy Under Uncertainty in Micro-Founded

Macroeconometric Models,� by A. Levin, A. Onatski, J. Williams and N. Williams,� in NBER

Macroeconomics Annual, ed. by M. Gertler, and K. Rogo¤. MIT Press, Cambridge, MA.

Woodford, M. (2003): Interest and Prices: Foundations of a Theory of Monetary Policy. Prince-

ton University Press, Princeton, NJ.


Federal Reserve Board

Northwestern University, NBER and CEPR

Coefficient Description Density 1/

Mean Std Median Std [ 5 , 95 ] Median Std [ 5 , 95 ]

ι p Price indexation B 0.50 0.15 0.84 0.04 [ 0.77 , 0.91 ] 0.83 0.05 [ 0.75 , 0.91 ]

ι w Wage indexation B 0.50 0.15 0.09 0.03 [ 0.05 , 0.14 ] 0.08 0.03 [ 0.04 , 0.13 ]

γ SS technology growth rate N 0.50 0.03 0.43 0.02 [ 0.39 , 0.47 ] 0.43 0.02 [ 0.39 , 0.47 ]

h Consumption habit B 0.50 0.10 0.81 0.03 [ 0.76 , 0.86 ] 0.84 0.03 [ 0.79 , 0.88 ]

λ p SS mark-up goods prices N 0.15 0.05 0.22 0.04 [ 0.16 , 0.28 ] 0.23 0.04 [ 0.17 , 0.29 ]

λ w SS mark-up wages N 0.15 0.05 0.17 0.04 [ 0.10 , 0.25 ] 0.16 0.04 [ 0.09 , 0.24 ]

L ss (log) SS labor N 396.83 0.50 397.10 0.46 [ 396.32 , 397.83 ] 397.01 0.49 [ 396.20 , 397.80 ]

π SS quarterly inflation N 0.50 0.10 0.56 0.10 [ 0.40 , 0.71 ] 0.55 0.10 [ 0.39 , 0.71 ]

r SS real interest rate N 0.50 0.10 1.03 0.07 [ 0.91 , 1.15 ] 1.03 0.07 [ 0.90 , 1.15 ]

ν Inverse Frisch labor G 2.00 0.75 1.59 0.35 [ 0.98 , 2.12 ] 1.59 0.48 [ 0.94 , 2.47 ]

ξ p Calvo prices B 0.75 0.10 0.90 0.01 [ 0.88 , 0.92 ] 0.91 0.01 [ 0.89 , 0.93 ]

ξ w Calvo wages B 0.75 0.10 0.61 0.05 [ 0.52 , 0.69 ] 0.66 0.05 [ 0.57 , 0.74 ]

χ Elasticity capital utilization costs G 5.00 1.00 6.90 1.10 [ 5.25 , 8.91 ] 7.13 1.09 [ 5.45 , 9.02 ]

S'' Investment adjustment costs G 3.00 0.75 2.72 0.48 [ 1.99 , 3.61 ] 3.30 0.57 [ 2.42 , 4.29 ]

Φ p Taylor rule inflation N 1.70 0.30 1.92 0.13 [ 1.71 , 2.15 ] 1.90 0.14 [ 1.67 , 2.14 ]

Φ y Taylor rule output G 0.13 0.10 0.10 0.02 [ 0.07 , 0.13 ] 0.08 0.02 [ 0.06 , 0.11 ]

( table continues on the next page )

Table 1: Prior densities and posterior estimates for the time invariant model and baseline model with stochastic volatility

Prior Posterior Time Invariant 2/ Posterior with Stochastic Volatility 3/

Coefficient Description Density 1/

Mean Std Median Std [ 5 , 95 ] Median Std [ 5 , 95 ]

Table 1: Prior densities and posterior estimates for the time invariant model and baseline model with stochastic volatility

Prior Posterior Time Invariant 2/ Posterior with Stochastic Volatility 3/

ρ R Taylor-rule B 0.60 0.20 0.81 0.02 [ 0.78 , 0.84 ] 0.84 0.02 [ 0.80 , 0.87 ]

ρ z Technology growth B 0.40 0.20 0.28 0.06 [ 0.18 , 0.39 ] 0.32 0.06 [ 0.21 , 0.43 ]

ρ g Government spending B 0.60 0.20 0.98 0.00 [ 0.98 , 0.98 ] 0.98 0.00 [ 0.98 , 0.98 ]

ρ μ Investment specific B 0.60 0.20 0.87 0.03 [ 0.82 , 0.92 ] 0.92 0.02 [ 0.88 , 0.96 ]

ρ φ Labor disutility B 0.60 0.20 0.90 0.03 [ 0.86 , 0.95 ] 0.89 0.04 [ 0.81 , 0.95 ]

ρ b Intertemporal preference B 0.60 0.20 0.84 0.05 [ 0.74 , 0.89 ] 0.83 0.05 [ 0.73 , 0.89 ]

σ r Monetary policy I 0.15 0.15 0.25 0.01 [ 0.23 , 0.28 ]

σ z Technology growth I 2.00 2.00 1.10 0.06 [ 1.01 , 1.20 ]

σ g Government spending I 1.00 2.00 0.55 0.03 [ 0.51 , 0.61 ]

σ μInvestment specific technology I 2.00 2.00 5.46 0.70 [ 4.43 , 6.76 ]

σ λ Mark-up I 0.15 0.15 0.17 0.01 [ 0.15 , 0.18 ]

σ φ Labor disutility I 4.00 2.00 10.41 1.11 [ 8.96 , 12.77 ]

σ b Intertemporal preference I 2.00 2.00 3.13 0.38 [ 2.67 , 3.98 ]

2/ Median, standard deviations and posterior percentiles from four chains of 140,000 draws each from the Random Walk metropolis algorithm initialized from dispersed starting values around the mode. We disrcard the initial 40,000 draws.

1/ N stands for Normal, B Beta, G Gamma and I Inverted-Gamma1 distribution

(log) Likelihood at median -1891.7 -1675.97

Calibrated coefficients: capital share (α) at 0.3, depreciation rate (δ) is 0.025, g at 1/0.77 which implies a SS government share of 0.22, and persistence of mark-up shocks (ρλ) set at zero. Relative to the text, γ corresponds to a quarterly growth rate in the estimation and is therefore multiplied by 100. Meanwhile, π and R are expressed as gross rates and multplied by 100 as well. Finally, the standard deviations of the innovations are also scaled by 100 for the estimation. All these changes are reflected in the specification of the priors.

3/ Median, standard deviations and posterior percentiles from the Random Walk Metropolis within Gibbs algorithm for the model with stochastic volatility. Results based on 3 chains of 150,000 draws each where we discarded the initial 50,000 and retain one in every 5 simulations from the remaining 100,000 draws.

Coefficient Description Median Std [ 5 , 95 ]

ι p Price indexation 0.86 0.04 [ 0.79 , 0.93 ]

ι w Wage indexation 0.11 0.03 [ 0.06 , 0.17 ]

γ SS technology growth rate 0.44 0.03 [ 0.39 , 0.48 ]

h Consumption habit 0.80 0.03 [ 0.76 , 0.84 ]

λ p SS mark-up goods prices 0.27 0.03 [ 0.22 , 0.33 ]

λ w SS mark-up wages 0.19 0.04 [ 0.13 , 0.26 ]

L ss (log) SS labor 396.64 0.51 [ 395.83 , 397.50 ]

π SS quarterly inflation 0.62 0.09 [ 0.46 , 0.77 ]

r SS real interest rate 1.04 0.08 [ 0.93 , 1.18 ]

ν Inverse Frisch labor 1.61 0.37 [ 1.13 , 2.35 ]

ξ p Calvo prices 0.94 0.03 [ 0.89 , 0.97 ]

ξ w Calvo wages 0.40 0.07 [ 0.31 , 0.52 ]

χ Elasticity capital utilization costs 6.88 1.00 [ 5.41 , 8.70 ]

S'' Investment adjustment costs 3.37 0.49 [ 2.57 , 4.19 ]

Φ p Taylor rule inflation 1.61 0.27 [ 1.27 , 2.13 ]

Φ y Taylor rule output gap 0.23 0.09 [ 0.06 , 0.36 ]

ρ R Taylor-rule 0.85 0.02 [ 0.82 , 0.88 ]

ρ z Technology growth 0.26 0.06 [ 0.16 , 0.36 ]

ρ g Government spending 0.98 0.00 [ 0.98 , 0.98 ]

ρ μInvestment specific technology 0.90 0.02 [ 0.86 , 0.94 ]

ρ φ Labor disutility 0.94 0.03 [ 0.89 , 0.97 ]

ρ b Intertemporal preference 0.85 0.04 [ 0.77 , 0.91 ]

-1698.12

Posterior

Table 2: Posterior estimates for stochastic volatility model with Taylor-rule responding to the output gap

(log) Likelihood at median

Priors and calibration identical to those reported for the stochastic volatility baseline model in Table 1.

Specification Log Marginal /1

Baseline stochastic volatility model 2/ -1824.6

Time Invariant Model /3 -1984.7

Determinate model with a single jump in volatilities -1925.8

Split model with a jump in all coefficients and active policy in first subsample -1947.9

Split model with a jump in all coefficients and passive policy in first subsample /4

-1959.0

Split model with a jump in volatilities and passive policy in first subsample -1941.1

Split model with a jump in policy only (passive to active) -1983.4

Stochastic volatility with Taylor-rule responding to output gap 5/

-1843.6

Full set of parameter estimates for the remaining models is available from the authors upon request

Table 3: Log-Marginal Data Densities for baseline stochastic volatility model and alternative specifications

Notes:

2/ Parameter estimates shown in column 3 of Table 1

/1 Log Marginal data density computed using the output of the MCMC simulators as described in Appendix A. Model favored by the data attains the highest marginal data density.

5/ Parameter estimates shown in Table 24/ Parameter estimates shown in Table 4 3/ Parameter estimates shown in column 2 of Table 1

Coefficient Description Median Std [ 5 , 95 ] Median Std [ 5 , 95 ]

ι p Price indexation 0.55 0.07 [ 0.45 , 0.67 ] 0.66 0.08 [ 0.51 0.79 ]

ι w Wage indexation 0.05 0.02 [ 0.02 , 0.09 ] 0.31 0.07 [ 0.20 0.43 ]

γ SS technology growth rate 0.48 0.02 [ 0.44 , 0.52 ] 0.47 0.02 [ 0.43 0.51 ]

h Consumption habit 0.67 0.03 [ 0.61 , 0.72 ] 0.77 0.04 [ 0.71 0.83 ]

λ p SS mark-up goods prices 0.23 0.04 [ 0.16 , 0.29 ] 0.18 0.04 [ 0.11 0.24 ]

λ w SS mark-up wages 0.11 0.04 [ 0.07 , 0.20 ] 0.19 0.04 [ 0.12 0.26 ]

L ss (log) SS labor 396.43 0.20 [ 396.08 , 396.74 ] 396.28 0.38 [ 395.62 396.88 ]

π SS quarterly inflation 0.52 0.08 [ 0.40 , 0.66 ] 0.82 0.07 [ 0.71 0.95 ]

r SS real interest rate 0.87 0.06 [ 0.77 , 0.98 ] 0.75 0.07 [ 0.64 0.86 ]

ν Inverse Frisch labor 0.58 0.12 [ 0.38 , 0.80 ] 1.92 0.61 [ 1.15 3.11 ]

ξ p Calvo prices 0.91 0.02 [ 0.88 , 0.94 ] 0.90 0.02 [ 0.88 0.93 ]

ξ w Calvo wages 0.69 0.04 [ 0.62 , 0.76 ] 0.45 0.08 [ 0.34 0.60 ]

χ Elasticity capital utilization costs

6.92 0.76 [ 5.73 , 8.15 ] 4.94 0.90 [ 3.70 6.63 ]

S'' Investment adjustment costs

1.48 0.19 [ 1.12 , 1.77 ] 2.83 0.53 [ 2.09 3.83 ]

Φ p Taylor rule inflation 0.52 0.09 [ 0.37 , 0.68 ] 2.37 0.19 [ 2.08 2.70 ]

Φ y Taylor rule output 0.19 0.03 [ 0.14 , 0.24 ] 0.02 0.01 [ 0.00 0.04 ]

( table continues on the next page )

Posterior Sample I: 1954q3-1983q4 /2

Posterior Sample II: 1984q1-2004q4 /2

Table 4: Posterior estimates for split model on two subsamples allowing for indeterminacy ( jump in all coefficients )

Coefficient Description Median Std [ 5 , 95 ] Median Std [ 5 , 95 ]

Posterior Sample I: 1954q3-1983q4 /2

Posterior Sample II: 1984q1-2004q4 /2

Table 4: Posterior estimates for split model on two subsamples allowing for indeterminacy ( jump in all coefficients )

ρ R Taylor-rule 0.69 0.04 [ 0.62 , 0.74 ] 0.84 0.02 [ 0.80 0.87 ]

ρ z Technology growth 0.18 0.05 [ 0.10 , 0.26 ] 0.31 0.08 [ 0.16 0.44 ]

ρ g Government spending 0.97 0.01 [ 0.94 , 0.98 ] 0.98 0.00 [ 0.98 0.98 ]

ρ μInvestment specific technology 0.66 0.05 [ 0.58 , 0.74 ] 0.92 0.02 [ 0.89 0.95 ]

ρ φ Labor disutility 0.07 0.02 [ 0.03 , 0.11 ] 0.93 0.05 [ 0.83 0.97 ]

ρ b Intertemporal preference 0.93 0.02 [ 0.89 , 0.95 ] 0.83 0.07 [ 0.69 0.92 ]

σ r Monetary policy 0.27 0.02 [ 0.25 , 0.31 ] 0.15 0.01 [ 0.13 0.17 ]

σ z Technology growth 1.27 0.08 [ 1.14 , 1.40 ] 0.80 0.07 [ 0.70 0.92 ]

σ g Government spending 0.71 0.05 [ 0.64 , 0.80 ] 0.48 0.04 [ 0.42 0.55 ]

σ μ Investment specific 5.08 0.43 [ 4.26 , 5.68 ] 2.63 0.37 [ 2.10 3.32 ]

σ λ Mark-up 0.19 0.01 [ 0.17 , 0.22 ] 0.13 0.01 [ 0.12 0.16 ]

σ φ Labor disutility 30.58 2.26 [ 26.59 , 34.25 ] 7.15 4.90 [ 4.29 16.24 ]

σ b Intertemporal preference 2.82 0.12 [ 2.60 , 2.99 ] 2.17 0.40 [ 1.69 2.98 ]

σ S 1/ Sunspot 0.08 0.01 [ 0.06 , 0.09 ]

M R 1/ -0.20 0.19 [ -0.51 , 0.12 ]

M z -0.10 0.10 [ -0.26 , 0.08 ]

M g -0.15 0.16 [ -0.39 , 0.15 ]

M μ -0.04 0.05 [ -0.12 , 0.02 ]

M λ -0.76 0.19 [ -1.02 , -0.43 ]

M φ -0.03 0.01 [ -0.04 , -0.02 ]

M b -0.23 0.07 [ -0.34 , -0.11 ]]

-1776.0

Notes:

(log) Likelihood at median

Calibrated parameters and priors for both subsamples are identical to those reported for the time invariant model in Table 1, with the exception of the coefficient on inflation in the Taylor rule. For that coefficient, given that we allow for indeterminacy, the prior in the first subsample is a Gamma with mean 1.1 and std 0.5, following Lubik and Schorfheide (2004). For the second subsample we return to the original normal prior centered at 1.7 with dispersion 0.3, that is used throught the paper.

2/ Median, standard deviations and posterior percentiles of 150,000 draws from the Random Walk Metropolis algorithm for the joint estimation of the model with indeterminacy in the first subsample. We discard the initial 50,000 draws.

1/ Indeterminacy results in eight new coefficients: the seven M parameters that modify the transmission of the fundamental shocks and the standard deviation of the sunspot shock (σS). As in Lubik and Schorheide (op cit) we center our prior in the first subsample on the continuity solution and therefore specify a prior for each M as Normal(0,0.5). See Appendix C for details.For the sunspot, meanwhile, we choose an Inverse Gamma 1 prior with mean and dispersion equal to 0.15.

A B C D

Output Median and [5,95]

posterior bands [ 0.66 , 0.88 ] [ 1.58 , 2.17 ] [ 0.87 , 1.28 ] [ 1.35 , 3.14 ]

Inflation Median and [5,95]

posterior bands [ 0.17 , 0.34 ] [ 0.17 , 0.32 ] [ 0.19 , 0.37 ] [ 0.36 , 1.00 ]

A B C D

Output Median and [5,95]

posterior bands [ 0.49 , 0.66 ] [ 0.98 , 1.19 ] [ 0.90 , 1.44 ] [ 0.97 , 1.78 ]

Inflation Median and [5,95]

posterior bands [ 0.40 , 1.16 ] [ 0.54 , 0.88 ] [ 0.50 , 0.99 ] [ 0.71 , 3.29 ]

Counterfactual: Monetary Policy

1.81

0.25

0.76

0.25

Estimated

Counterfactual: All coefficients

except volatilities

2.09

0.58

PANEL a. Passive policy in first subsample

Table 5: Counterfactual standard deviations (std) of output and inflation

PANEL b. Active policy in first subsample

Estimated Counterfactual: Monetary Policy


and private sector coefficients

Counterfactual: All coefficients

except volatilities


and private sector coefficients

1.04

0.28

Ratios of model implied stds (either estimated in the second subsample or counterfactual) to the estimated std in the first subsample. First subsample ends in 1983q4, second subsample starts in 1984q1. Medians and [5,95] posterior probabilities computed using MCMC draws for the models in which all coefficients are allowed to change in 1984q1. Panel a: monetary policy is passive and equlibrium indeterminate in first subsample. Panel b: monetary policy is active in first subsample. Column A: ratio of stds implied by the estimated models in the second (numerator) to the first (denominator) subsamples. Column B: ratio of counterfactual std when replace in the first subsample Taylor rule coefficients estimated in the second subsample. Column C: same as column B, but in addition replace private sector coefficients with those estimated in the second subsample. Column D: same as column C, but in addition replace the autocorrelation coefficients of the shocks with those estimated in the second subsample.

0.69 0.73 0.70 1.40

0.57 1.07 1.13 1.31

Variances Interval

lengths (90 percent)

Batched Means

Autoregressive Spectral Estimate

ι p Price indexation 1.00 1.00 0.21 0.23

ι w Wage indexation 1.00 1.00 0.16 0.15

γ SS technology growth rate 1.00 1.00 0.92 0.91

h Consumption habit 1.00 1.00 0.07 0.06

λ p SS mark-up goods prices 1.00 1.00 0.17 0.20

λ w SS mark-up wages 1.01 1.01 0.67 0.68

L ss (log) SS labor 1.00 1.00 0.29 0.29

π SS quarterly inflation 1.00 1.00 0.44 0.46

r SS real interest rate 1.01 1.01 0.03 0.03

ν Inverse Frisch labor 1.00 1.00 0.41 0.38

ξ p Calvo prices 1.00 1.00 0.44 0.49

ξ w Calvo wages 1.01 1.00 0.84 0.83

χElasticity capital utilization costs 1.00 1.00 0.12 0.09

S'' Investment adjustment costs 1.00 1.00 0.86 0.86

Φ p Taylor rule inflation 1.00 1.00 0.40 0.42

Φ y Taylor rule output 1.00 1.00 0.97 0.97

ρ R Taylor-rule 1.00 1.00 0.98 0.98

ρ z Technology growth 1.00 1.00 0.32 0.29

ρ g Government spending 1.00 1.00 0.25 0.25

ρ μ Investment specific 1.00 1.00 0.42 0.44

ρ φ Labor disutility 1.02 1.01 0.84 0.85

ρ b Intertemporal preference 1.00 1.00 0.64 0.65

1/ Computed for draws of the baseline stochastic volatility model (posterior estimates reported in Table 1)

3/ P-values for the null hypothesis of equal means based on the initial 20 percent and the latter half of the draws when pooling all individual chains. See Geweke (1992) for details. We compute the corresponding Wald-statistics and associated p-values using batched means (Chib, 2001) and autoregressive spectral estimates for the variance of the (serially correlated) draws. These alternative estimates of the long-run variance yield an overall identical picture regarding the equality of means for the pooled chain.

Table 6: Convergence Diagnostics for the Baseline Stochastic Volatility Model /1

2/ Potential scale reduction factors for multiple chains below 1.2 or 1.1 are regarded as indicative of convergence. The first column reports PSRF using within and between variances. The second column PSRF based on empirical 90 percent interval lengths (length of total-sequence interval to mean length of within-sequence intervals), as proposed by Brooks and Gelman (1998)

Potential Scale Reduction Factors (PSRF) /2

P-values for Geweke's tests of equality of means

/3

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B. Neutral Technology

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C. Government Spending

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D. Investment Specific

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E. Price Mark-Up

0

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50

60

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F. Labor Disutility

2

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4

5

6

7

60 65 70 75 80 85 90 95 00

G. Intertemporal Preference

Figure 1: Stochastic Volatility of Each Shock in DSGE Model

Median and 5-95 percentiles for the time varying volatility of each disturbance computedwith the draws generated in the estimation of the baseline stochastic volatility model

.0

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Figure 2. Variance Decomposition for Output Growth

Median share and associated 5-95 percentiles for each disturbance computed with the draws generatedin the estimation of the baseline stochastic volatility model

.2

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.8

55 60 65 70 75 80 85 90 95 00 05

A. Consumption, Intertemporal Preference

.0

.1

.2

.3

.4

.5

.6

.7

.8

55 60 65 70 75 80 85 90 95 00 05

B. Federal Funds Rate, Monetary Policy

.0

.1

.2

.3

.4

.5

.6

55 60 65 70 75 80 85 90 95 00 05

C. Inflation, Price Mark-up

.0

.1

.2

.3

.4

.5

.6

.7

.8

.9

55 60 65 70 75 80 85 90 95 00 05

D. Inflation, Investment Specific

0.0

0.2

0.4

0.6

0.8

1.0

55 60 65 70 75 80 85 90 95 00 05

E. Inflation, Labor Disutility

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

55 60 65 70 75 80 85 90 95 00 05

F. Investment, Investment Specific

0.0

0.2

0.4

0.6

0.8

1.0

55 60 65 70 75 80 85 90 95 00 05

G. Labor, Labor Disutility

.0

.1

.2

.3

.4

.5

.6

55 60 65 70 75 80 85 90 95 00 05

H: Real Wages, Neutral Technology

Figure 3: Selected Variance Decomposition for Other Series( Series, Shock )

Median share and associated 5-95 percentiles for each disturbance computedwith the draws generated in the estimation of the baseline stochastic volatility model

.0

.1

.2

.3

.4

.5

55 60 65 70 75 80 85 90 95 00 05

A. Monetary Policy

0.0

0.2

0.4

0.6

0.8

1.0

55 60 65 70 75 80 85 90 95 00 05


.0

.1

.2

.3

.4

.5

.6

.7

55 60 65 70 75 80 85 90 95 00 05


0.0

0.2

0.4

0.6

0.8

1.0

55 60 65 70 75 80 85 90 95 00 05


.0

.1

.2

.3

.4

.5

55 60 65 70 75 80 85 90 95 00 05

E. Price Mark-Up

0.0

0.2

0.4

0.6

0.8

1.0

55 60 65 70 75 80 85 90 95 00 05

F. Labor Disutility

.0

.1

.2

.3

.4

55 60 65 70 75 80 85 90 95 00 05


Figure 4: Spectral Decomposition for Output

Spectral decomposition for periodicities between 8 and 32 quarters forthe level of output in deviation from the model implied stochastic trend.

Median share and associated 5-95 percentiles for each disturbancecomputed with the draws generated in the estimation of the baseline

stochastic volatility model.

1.0

1.5

2.0

2.5

3.0

3.5

55 60 65 70 75 80 85 90 95 00

Figure 5: Time Varying Volatility of Output Growth Implied by Baseline DSGE Model

Median and associated 5-95 percentiles computed with the draws generatedin the estimation of the baseline stochastic volatility model

0.5

1.0

1.5

2.0

2.5

3.0

3.5

82 84 86 88 90 92 94 96 98 00 02 04

A. Monetary Policy

0.8

1.2

1.6

2.0

2.4

2.8

3.2

3.6

82 84 86 88 90 92 94 96 98 00 02 04


0.8

1.2

1.6

2.0

2.4

2.8

3.2

3.6

82 84 86 88 90 92 94 96 98 00 02 04


1.0

1.5

2.0

2.5

3.0

3.5

82 84 86 88 90 92 94 96 98 00 02 04


0.8

1.2

1.6

2.0

2.4

2.8

3.2

3.6

82 84 86 88 90 92 94 96 98 00 02 04

E. Price Mark-Up

0.8

1.2

1.6

2.0

2.4

2.8

3.2

3.6

82 84 86 88 90 92 94 96 98 00 02 04

F. Labor Disutility

0.8

1.2

1.6

2.0

2.4

2.8

3.2

3.6

82 84 86 88 90 92 94 96 98 00 02 04

Actual std Counterfactual Std 5 and 95 posterior bands


Figure 6: Actual and Counterfactual Standard Deviation (Std) for Output Growth

Counterfactual std obtained by fixing for the remainder of the sample the std of each shock, one at a time,at the 4 quarter average level of that shock's time varying standard deviation in 1980. These are computed

with the draws generated in the estimation of the baseline stochastic volatility model

1

2

3

4

5

6

7

82 84 86 88 90 92 94 96 98 00 02 04

A. Monetary Policy

1

2

3

4

5

6

7

82 84 86 88 90 92 94 96 98 00 02 04


1

2

3

4

5

6

7

82 84 86 88 90 92 94 96 98 00 02 04


1

2

3

4

5

6

7

82 84 86 88 90 92 94 96 98 00 02 04


1

2

3

4

5

6

7

82 84 86 88 90 92 94 96 98 00 02 04

E. Price Mark-Up

1

2

3

4

5

6

7

82 84 86 88 90 92 94 96 98 00 02 04

F. Labor Disutility

1

2

3

4

5

6

7

82 84 86 88 90 92 94 96 98 00 02 04

Actual Std Counterfactual Std 5 and 95 posterior bands


Figure 7: Actual and Counterfactual Standard Deviation (Std) for Inflation

Counterfactual std obtained by fixing for the remainder of the sample the std of each shock, one at a time,at the 4 quarter average level of that shock's time varying standard deviation in 1980. These are computed

with the draws generated in the estimation of the baseline stochastic volatility model

Figure 8: Time varying volatility of GDP growth (solid blue line, left axis) and of the growth rate of the relative price of investment (dashed red line, right axis)

1965 1970 1975 1980 1985 1990 1995 2000 20050.3

0.5

0.7

0.9

1.1

1.3

1.5

1965 1970 1975 1980 1985 1990 1995 2000 20050.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Obtained using a (non-centered) 10-year moving window

Documents

The Time Varying Volatility of Macroeconomic Fluctuations